Market-Based Corrective Actions - Finance...

Market-Based Corrective Actions

Philip BondUniversity of Pennsylvania

Itay GoldsteinUniversity of Pennsylvania

Edward Simpson PrescottFederal Reserve Bank of Richmond

Many economic agents take corrective actions based on information inferred from marketprices of firms’ securities. Examples include directors and activists intervening in themanagement of firms and bank supervisors taking actions to improve the health of financialinstitutions. We provide an equilibrium analysis of such situations in light of a key problem:if agents use market prices when deciding on corrective actions, prices adjust to reflect thisuse and potentially become less revealing. We show that market information and agents’information are complementary, and discuss measures that can increase agents’ ability tolearn from market prices. (JEL D53, D80, G14, G21, G28, G34)

An established view in financial economics is that financial-market prices pro-vide useful and important information about firms’ fundamentals. The idea,going back to Hayek (1945), is that financial markets collect the private in-formation and beliefs of many different people who trade in firms’ securitiesand hence provide an efficient mechanism for information production andaggregation. A large body of empirical evidence demonstrates the ability offinancial markets to produce information that accurately predicts future events.One of the most cited examples is provided by Roll (1984), who suggeststhat orange juice futures predict the weather better than the National WeatherService.

We thank Beth Allen, Franklin Allen, Mitchell Berlin, Alon Brav, Thomas Chemmanur, Douglas Diamond, AlexEdmans, Andrea Eisfeldt, Gary Gorton, Wei Jiang, Richard Kihlstrom, Rajdeep Sengupta, Holger Spamann,Annette Vissing-Jorgensen, an anonymous referee, and the editor (Paolo Fulghieri) for their comments andsuggestions. We also thank seminar participants at numerous universities and conferences for their commentsand suggestions. The views expressed in this paper do not necessarily reflect the views of the Federal ReserveBank of Richmond or the Federal Reserve System. Send correspondence to Philip Bond, Wharton School,University of Pennsylvania, 2300 Steinberg Hall-Dietrich Hall, 3620 Locust Walk, Philadelphia, PA 19104;telephone: (215) 898-2370; fax: 215-898-6200. E-mail: [email protected].

C© The Author 2009. Published by Oxford University Press on behalf of The Society for Financial Studies.All rights reserved. For Permissions, please e-mail: [email protected]:10.1093/rfs/hhp059 Advance Access ublication September 20, 2009p

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Given this basic premise, it is not surprising that many economic agentstake actions (or are encouraged to take actions) driven by the informationthat is summarized in market prices. In corporate governance, it is widelybelieved that low market valuations trigger the replacement of CEOs by theboard of directors or attract various actions by shareholder activists. In banksupervision, regulators are frequently encouraged to learn from market pricesof bank securities before making an intervention decision. Even corporatemanagers are believed to be influenced by market prices of their firms’ securitieswhen making a decision to invest or acquire another firm.

Our article deals with a fundamental theoretical issue that needs to be con-sidered when market-based actions are discussed or advocated. Since marketprices are forward looking, they reflect information not only about firms’ fun-damentals, but also about the resulting actions of various agents (i.e., directors,activists, regulators, or managers). In some cases, this considerably compli-cates the inference of information from the price. Let us consider the exampleof a board of directors that is deciding whether to replace a CEO. If the boardknows that the CEO is of low quality, they will replace him. This correctiveaction will benefit the shareholders of the firm and thus increase the price of itsshares. So inferring information from the price about the quality of the CEO isa challenge: a moderate price may indicate either that the CEO is bad and thatthe board is expected to intervene and replace him, or that the CEO is not badenough to justify intervention.

We provide a theoretical analysis of such a situation in a general framework.Specifically, we characterize the rational expectations equilibria of a modelin which the price of a firm’s security both affects and reflects the decisionof an agent on whether to take an action that affects the value of the firm.Our focus is on the theoretically challenging, yet empirically relevant, casedescribed above—i.e., where the price exhibits nonmonotonicity with respectto the fundamentals due to the positive effect that the agent’s corrective action(taken when the fundamentals are bad) has on the value of the firm’s security.In this case, learning from the price is complicated by the fact that two ormore fundamentals may be associated with the same price. The equilibriumanalysis, in turn, becomes quite challenging given that the price has to reflectthe expected action, which depends on the price in a nontrivial way.

Before describing the results of our analysis, let us explain the relationbetween our model and the existing literature. A key feature of our analysisis that prices in financial markets affect the real value of securities via theinformation they provide to decision makers. In this, our model is differentfrom the vast majority of papers on financial markets, where the real valueof securities is assumed to be exogenous (e.g., Grossman and Stiglitz 1980).Our article contributes to a growing literature that analyzes models in whichan economic agent seeks to glean information from a market price and thentakes an action that affects the value of the security—see Fishman and Hagerty(1992); Khanna, Slezak, and Bradley (1994); Boot and Thakor (1997); Dow and

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Gorton (1997); Fulghieri and Lukin (2001); Goldstein and Guembel (2008);Bond and Eraslan (2008); and Dow, Goldstein, and Guembel (2008).1

The above papers, however, do not consider the main case of interest in ourmodel, where the price function is nonmonotone with respect to the fundamen-tals and inference from the price is complicated by the fact that one price can beconsistent with two or more fundamentals. Hence, all these other papers relateto a special case of our model, the analysis of which is summarized in Section2.3, where the price function is monotone.

Perhaps the only theoretical mention of the problem we focus on here ismade by Bernanke and Woodford (1997) in the context of monetary policy. Theyobserve that if the government tries to implement a monetary policy that is basedon inflation forecasts, a possible consequence is the nonexistence of rationalexpectations equilibria.2 Our analysis goes much beyond this basic observation.In particular, by studying a richer model, we are able to demonstrate under whatconditions an equilibrium exists and to characterize the informativeness of theprice and the efficiency of the resulting corrective action when an equilibriumdoes exist. Thus, we make a first step in analyzing the equilibrium results ofa very involved problem, where the use of market data is self-defeating inthe sense that the reflection of the expected market-based action in the pricedestroys the informational content of the price.

Turning to the results of our equilibrium analysis, we show that a key pa-rameter in the characterization of equilibrium outcomes is the quality of theinformation held by the agent making the corrective-action decision. Whenthe agent has relatively precise information, he3 is able to learn from marketprices and implement his preferred intervention rule as a unique equilibrium.When the agent’s information is moderately precise, additional equilibria existin which the agent misinterprets the market and intervenes either too muchor too little. Interestingly, in this range, the type of equilibrium—i.e., whetherthere is too much or too little intervention—depends on whether the tradedsecurity has a convex payoff (equity) or a concave payoff (debt). Finally, anagent whose information is imprecise cannot learn from the market and socannot implement his preferred intervention strategy in equilibrium.

Our analysis generates several normative implications for market-based cor-rective actions. First and foremost, we demonstrate that there is a strong com-plementarity between an agent’s direct sources of information and his use of

1 See also Subrahmanyam and Titman (1999) and Foucault and Gehrig (2008) for models in which the informationin the price affects a corporate action, but this is not reflected in the price of the security; see Ozdenoren andYuan (2008) for a model in which prices affect real values in an exogenously specified way. Also related arepapers in which a feedback loop exists between market prices and firm values due to the presence of market-based compensation contracts (e.g., Holmstrom and Tirole 1993; Admati and Pfleiderer, forthcoming; Edmans,forthcoming).

2 For a similar observation in the context of bank supervision, see the recent working paper by Birchler andFacchinetti (2007).

3 Throughout the article, we refer to the agent as male, although, of course, the agent could be female, or not aperson (e.g., the government).

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market data. An agent’s direct sources of information are crucial for the effi-cient use of market data. This implication is derived despite the fact that ourmodel endows the market with perfect information about the fundamentals.The role of the agent’s own information in our model is thus to enable him totell the extent to which the price reflects fundamentals as opposed to expec-tations about the agent’s own action. Second, we analyze other measures thathelp the agent implement his preferred market-based intervention policy, evenwhen the information gap between the market and the agent is not small. Thesemeasures include tracking the prices of multiple traded securities, revealing theagent’s information (transparency or disclosure), and introducing a securitythat pays off in the event that the agent takes a corrective action (a predictionmarket).

Our article also offers several positive implications. Our leading applicationshave been the subject of wide empirical research trying to detect the relationbetween market prices and the resulting actions. Our article suggests that thequality of information of agents outside the financial market and the shapeof the security, whose price is observed, are key factors affecting the relationbetween the price and the resulting action. In addition, we argue that two keyfeatures of our theory have to be taken into account in empirical researchon market-based intervention. First, if agents use the market price in theirintervention decision, there will be dual causality between market prices andthe intervention decision. In the context of shareholder activism in closed-endfunds, Bradley et al. (2009) conduct empirical analysis that takes into accountthis dual causality. Failing to account for the dual causality will produce resultsthat appear as just a weak relation between prices and actions. Second, whenthe information that agents have outside the financial market is not preciseenough, our model generates equilibrium indeterminacy, making the relationbetween market prices and intervention difficult to detect.

The remainder of the article is organized as follows. In Section 1, we presentthe general model. Section 2 characterizes equilibrium outcomes. Section 3 dis-cusses leading applications of the model. In Section 4, we consider robustnessissues and extensions of the basic model. Section 5 studies ways to improvethe efficiency of learning from the market. Section 6 concludes. All proofs arerelegated to Appendix A.

1. The model

The model has one firm, an agent, and a financial market that trades a securityof the firm. There are three dates, t = 0, 1, 2. At date 0, the price of the securityis determined in the market. At date 1, the agent, based on his information andthe information he gleans from the security price, may decide to take an action(intervene) that affects the value of the firm. At date 2, security holders arepaid. As we discuss in the introduction, this is a general framework that cancapture various situations where an agent seeks information from a security

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price in order to decide whether to take an action that ultimately affects thevalue of the security. In Section 3, we discuss in detail some leading applica-tions, including CEO replacement, shareholder activism, bank regulation, andcorporate investment.

1.1 The firmIn the absence of intervention, at date 2, the firm’s assets generate a gross cashflow y = θ + δ, where θ is drawn at date 0 from a distribution with density gand support (θ, θ),4 and δ is drawn at date 2 from a distribution with densityf . We refer to θ as the fundamental of the firm, while δ represents residualuncertainty. The residual δ is independent of the fundamental θ, and its meanE[δ] is equal to 0.

Different types of investors—including debt holders of different prioritiesand equity holders—have claims on the firm’s cash flows. In most of the article,we analyze a situation where the agent deciding whether to take the action atdate 1 learns from the date-0 price of one traded security. We let X (θ) denotethe value of this security absent intervention as a function of the realizedfundamental θ. Since most of our applications deal with agents learning fromthe price of the firm’s equity, we will be primarily interested in the value of thefirm’s equity. In this case,

X (θ) =∫

D−θ

(θ + δ − D) f (δ) dδ, (1)

where D is the face value of the firm’s outstanding debt. Since X ′(θ) =∫D−θ

f (δ) dδ > 0 and X ′′(θ) = f (D − θ) > 0, the value of equity X (θ) in-creases and is convex in the fundamental θ. The convex shape of the securitywill play a role in the characterization of equilibrium outcomes in the nextsection.5

1.2 The agentWe model the agent as having the opportunity to intervene in the firm’s businessat date 1. If the agent intervenes, the firm’s date-2 cash flow increases by T (θ).6

When T (θ) > 0, the intervention is a corrective action. We assume that T (θ)weakly decreases in θ. That is, the benefit from the agent’s intervention is highwhen the firm’s fundamentals are low. This is a natural assumption reflectingthe idea that there is more room for improvement when the state is bad. Still,

4 If the support is unbounded, θ = −∞ and/or θ = ∞.

5 In Section 4, we briefly describe how the results would change if the security that the agent learns from isconcave. This case is particularly relevant for bank supervision, since regulators are often advised to learn fromthe price of a bank’s debt. In Section 5, we allow for learning from both a concave and a convex security at thesame time.

6 Although we model intervention as a binary decision, we do allow for probabilistic intervention. However, sincewe also require that the agent’s decision is optimal given his information, probabilistic intervention rarely occurs.

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θ + T (θ) increases in θ—that is, in the presence of intervention, the totalexpected cash flow available to the firm increases in fundamentals.

Intervention by the agent affects the value of the security through its effecton the firm’s cash flows. Denoting the expected value of intervention for thesecurity holders as U (θ), we get

U (θ) = X (θ + T (θ)) − X (θ). (2)

We assume a fixed cost of intervention C > 0, which is borne by the agent.7

The benefit of intervention for the agent is denoted as V (θ). When decidingwhether to intervene, the agent weighs the private cost against the benefit.

For some applications, it is natural to consider the special case in which theagent internalizes the full effect of his action (i.e., V coincides with the effecton expected cash flow: V ≡ T ). However, our analysis is general enough tocover a range of other possibilities. The only assumption we make regarding theagent’s gain from intervention is that V (θ) ≥ C if and only if the fundamentallies below some unique cutoff, θ. That is, a fully informed agent would interveneonly when the fundamental is sufficiently low. For example, this would be thecase if V is a decreasing function, and V exceeds (is less than) C at very low(high) fundamentals.

1.3 Information and pricesA key point in our analysis is that the agent does not directly observe θ, butinstead must try to infer it from the market price of the firm’s security. Therealization of θ is known in the market at date 0 and serves as a basis for theprice formation. In particular, the price P(θ) is set to reflect the expected valueof the security given the fundamental θ (taking into account the probability ofintervention).

In addition to the market price, at date 0, the agent observes a noisy signal ofθ: φ = θ + ξ. The noise with which the agent observes the fundamental, ξ, isuniformly distributed over [−κ, κ], and φ is not observed by the market.8 Theagent’s intervention policy is then a probability of intervention I (P,φ) ∈ [0, 1],which is a function of the agent’s own signal φ and the observed price of thefirm’s security P .

One limitation of our information structure is that it assumes that the agentalways knows less than the information collectively possessed by market par-ticipants (i.e., the information of market participants aggregates to θ, while theagent only observes a noisy signal of θ). This assumption helps simplify the

7 Having a fixed cost C is not necessary for our analysis. The only thing that we will need is that C does notdecrease too fast in θ.

8 The general nature of the inference problem studied in the article does not depend on the assumption that thenoise in the agent’s signal is uniformly distributed. It depends only on having some noise in the agent’s signaland on the nonmonotonicity of the price with respect to the fundamentals (to be explained later). However, thedetails of the analysis do make use of the uniformity assumption.

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analysis and exposition in the article, without harming its main goal, whichis to analyze equilibrium outcomes when the agent learns from the market.In Section 4, we discuss the robustness of our model to this assumption andconsider an extension in which the agent sometimes has more information thanthe market.

2. Equilibrium analysis

2.1 Equilibrium definitionIn equilibrium, the price P(·) reflects the expected value of the security giventhe fundamental θ and the intervention probability (for a given interventionpolicy I (·, ·)). Formally, the following rational expectations equilibrium (REE)condition must hold:

P(θ) = X (θ) + Eφ[I (P(θ),φ)|θ]U (θ) for all θ. (3)

The first component in this expression is the expected value of the securityabsent intervention given the fundamental θ. The second component is theadditional value stemming from the possibility of intervention, the probabilityof which depends on the price P(·) and the agent’s own signal φ.

The second equilibrium condition is that the agent’s intervention decisionmaximizes his utility, given his beliefs about the fundamental θ. Specifically,the agent intervenes with probability 1 (respectively, 0) if the expected benefitfrom intervention is strictly greater (smaller) than the cost. Formally,9

I (P, φ) ={

1 if Eθ[V (θ)|P(θ) = P and φ] > C

0 if Eθ[V (θ)|P(θ) = P and φ] < C. (4)

With slight abuse of language, we will commonly refer to condition (4) as the“best response” condition.10 In Section 5.4, we analyze the model under thealternative assumption that the agent can commit to an intervention rule, andso condition (4) need not hold.

The formal definition of equilibrium is then as follows:

Definition 1. A pricing function P(·) and an intervention policy I (·, ·) to-gether constitute an equilibrium if they satisfy the REE condition (3) and thebest-response condition (4).

2.2 Agent-preferred equilibriaWe start by defining an important class of equilibria:

9 Note that the intervention probability can lie anywhere in [0, 1] if Eθ[V (θ)|P(θ) = P and φ] = C .

10 Recall that the market price is determined by the rational expectations condition rather than as the outcome of astrategic game.

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Definition 2. An agent-preferred equilibrium is one in which the agent in-tervenes if the benefit exceeds the cost, V (θ) > C , and does not intervene ifV (θ) < C .

Any equilibrium with fully revealing prices (i.e., each price is associatedwith one fundamental) is an agent-preferred equilibrium. Additionally, and aswe show below, there exist equilibria in which the price is not fully revealing,but in which the combination of the price and the agent’s own signal allowshim to fully infer the fundamental. Such equilibria also feature agent-preferredintervention.

From Equation (2), the price function for the security under the agent’spreferred intervention rule is given by

P(θ) ={

X (θ + T (θ)) if θ < θ

X (θ) if θ > θ. (5)

The main questions we are interested in are whether an agent-preferred equi-librium exists, and if it does, then whether it is the unique equilibrium outcome.

2.3 Monotone price function: T (θ) ≤ 0We start with a simple case where agent-preferred intervention is the uniqueequilibrium outcome, independent of the accuracy of the agent’s signal. Thishappens when intervention at θ reduces the firm’s expected cash flow—i.e.,T (θ) ≤ 0. A leading example in the context of bank supervision is a firesaleliquidation of bank assets. Here, the regulator liquidates in order to ensurepayment to depositors. This, however, reduces the cash flows to other claimholders and thus the value of their securities declines. The formal result for thiscase is in Proposition 1.

Proposition 1. If T (θ) ≤ 0, then (for all agent signal accuracies κ) an equi-librium with agent-preferred intervention exists, and is the unique equilibrium.

To see the intuition behind this result, it is useful to inspect Figure 1,which displays the price function (5) for this case.11 In the figure, we seethe price of the security under intervention—X (θ + T (θ))—and the price un-der no intervention—X (θ). The agent wishes to intervene if and only if θ < θ,and thus his preferred intervention generates a price function that is depictedby the bold lines in the figure. The key property of this function is that it ismonotone in θ. Hence, every level of the fundamental θ is associated with a dif-ferent price. This implies that the agent can learn the realization of θ preciselyfrom the price and thus act in his preferred way, regardless of how imprecisehis signal is.

11 Note that Figure 1 and the other figures in the article are only schematic. In particular, the functions are drawnas linear functions, although they need not be linear.

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ˆ

X(θ)

X(θ + T(θ))

P(θ)

θθ

Figure 1Security price under agent-preferred intervention when T (θ) < 0

X(θ)

X(θ + T(θ))P(θ)

ˆ ˆθ + T(θ) θθθ

Figure 2Security price under agent-preferred intervention when T (θ) > 0

This case of a monotone price function is the one analyzed in the existingliterature on the feedback effect from asset prices to the real value of securities(see the introduction). We now turn to the case that is the focus of our analysis—that of a nonmonotone price function.

2.4 Nonmonotone price function: T (θ) > 0In many situations, things are not as simple as described in the previous subsec-tion. In particular, consider any application in which intervention is beneficialfor the agent only if it increases expected cash flows (i.e., V (θ) > 0 only ifT (θ) > 0). That is, the agent would like to intervene so as to improve thefirm’s health. Since the agent’s benefit from intervention is equal to the agent’sprivate cost of intervention at the fundamental θ—i.e., V (θ) = C—it followsthat T (θ) > 0. At θ, intervention is a corrective action and is good for the firmvalue, but the agent is indifferent between intervening and not intervening dueto the private cost C that he has to bear.

For the remainder of the article, we focus on the case in which interventionis corrective at θ and below. Figure 2 displays the price function (5) for thiscase.

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The inspection of Figure 2 reveals the difficulty in obtaining an equilibriumwith agent-preferred intervention when T (θ) > 0. The agent’s preferred inter-vention rule is to intervene if and only if θ is below θ. As we can see in the figure,because T (θ) > 0, the price function under the preferred intervention rule isnonmonotone around θ. That is, as the fundamental decreases and crosses thethreshold θ, the agent wishes to intervene. Intervention, in turn, increases thevalue of the security from X (θ) to X (θ + T (θ)).12

The implication of this nonmonotonicity is that fundamentals on both sidesof θ have the same price. In particular, consider the interval of fundamentals[θ, θ + T (θ)] depicted in the figure. Here, θ is defined such that θ + T (θ) ≡ θ.The three fundamentals—θ, θ, and θ + T (θ)—are related to each other, as theexpected cash flow in the second (third) fundamental without intervention isthe same as the expected cash flow in the first (second) fundamental withintervention. The interval [θ, θ + T (θ)] can be separated into two subintervals:[θ, θ] and [θ, θ + T (θ)]. Under agent-preferred intervention, every fundamentalin [θ, θ] has a fundamental in [θ, θ + T (θ)] with which it shares the same price.This implies that the agent can infer neither the level of the fundamental, nor hispreferred action, from the price alone. Essentially, the fact that the price reflectsthe expected action of the agent (i.e., intervention below θ) makes learning fromthe price more difficult.

A natural conjecture that follows from this discussion is that the possibility ofachieving agent-preferred intervention in equilibrium depends on the precisionof the agent’s signal. A precise signal will enable the agent to distinguishbetween different fundamentals that have the same price. We provide an analysisof equilibrium outcomes based on the precision of the agent’s signal.

As we noted before, another important factor in determining equilibriumoutcomes is the shape of the value of the firm’s security with respect to thefundamentals. We focus the presentation on the results for a convex security,where both X (θ) and X (θ + T (θ)) are convex with respect to θ. Moreover,we assume that |T ′(θ)| is sufficiently small between θ − 2κ and θ + 2κ. Thisimplies that the benefit from intervention does not decrease very fast in thefundamental. Intuitively, this helps preserve the features implied by a convexsecurity by ensuring that U (θ) (defined as X (θ + T (θ)) − X (θ)) is increasing.We have also analyzed the model for the cases in which T (θ) decreases fastand/or the security is concave. We briefly discuss the results of this alternativeanalysis in Section 4.

The next proposition characterizes equilibrium outcomes under the aboveassumptions:

12 While in our model nonmonotonicity arises in part from the discreteness of the intervention decision, it isimportant to note that this feature is not necessary for nonmonotonicity. Indeed, Birchler and Facchinetti (2007)show that as long as there is some fixed cost in intervention, nonmonotonicity will be a feature of the pricefunction even if the intervention decision is continuous.

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Proposition 2. If κ < T (θ)/2, then there exists an equilibrium with agent-preferred intervention. This is the unique equilibrium if κ ≤ κ, for someκ ∈ (0, T (θ)/2), while for κ sufficiently close to T (θ)/2, there are addi-tional equilibria that do not exhibit agent-preferred intervention. Conversely,if κ > T (θ)/2, then there is no equilibrium with agent-preferred intervention,and if κ > T (θ − 2κ)/2, no equilibrium exists.

The proposition confirms that the precision of the agent’s signal is a crucialparameter in determining whether market-based corrective action can achievethe agent’s goal. When the precision is high (κ is low), the agent-preferred inter-vention is achieved as a unique equilibrium. When the precision is intermediate,there may also exist other equilibria in which agent-preferred intervention isnot achieved. We analyze equilibria of this sort in the next subsection; seeProposition 3. Finally, when the precision is low (κ is high), agent-preferredintervention cannot be achieved in equilibrium. We now give intuition for theseresults.

Why is agent-preferred intervention an equilibrium when κ < T (θ)/2? Un-der the agent-preferred intervention rule, there are at most two fundamentalsassociated with each price. Suppose that θ1 and θ2, θ1 < θ < θ2, have the sameprice. Under the agent’s preferred intervention rule, these fundamentals are ata distance T (θ1) from each other (see Figure 2). Since 2κ < T (θ) ≤ T (θ1), theagent’s signal enables him to tell these fundamentals apart when observing aprice that is consistent with both of them. Then, knowing the realization of thefundamental, the agent can follow his preferred intervention rule. Two pointsare worth stressing. First, in this equilibrium, both the price and the signal servean important role: the price tells the agent that one of two different fundamentalshas been realized, while the signal enables the agent to differentiate betweenthese two fundamentals. Second, the construction of this particular equilibriumrelies on the assumption that the distribution of the noise in the agent’s signalis bounded. Economically, this amounts to saying that the agent is able to ruleout some realizations of the fundamental after observing his own signal.13

While κ < T (θ)/2 guarantees the existence of an agent-preferred equilib-rium, there may exist other equilibria where the distance between fundamentalssharing the same price is smaller than 2κ, making inference from the price hardand leading to interventions that are different than the agent-preferred rule.However, when κ is sufficiently small, Proposition 2 rules out such equilib-ria. Although intuitive, the proof is long and involved. The key difficulty is theneed to rule out equilibria in which there are an infinite number of fundamentalsassociated with the same price.

Finally, when κ > T (θ)/2, agent-preferred intervention cannot occur in equi-librium. This is because in an equilibrium with agent-preferred intervention

13 The fact that the noise term ξ has bounded support is a direct consequence of the assumption that it is distributeduniformly. As noted in footnote 8, this assumption is needed for tractability.

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there are fundamentals at a distance of T (θ) from each other on both sides of θ

that have the same price. Since 2κ > T (θ), the signal does not enable the agentto always distinguish between two fundamentals that have the same price. Thus,given a price that is associated with two fundamentals, it is impossible for theagent to always intervene at one fundamental and never intervene at the other,and therefore agent-preferred intervention cannot occur.

The proposition states a stronger result for the range where κ > T (θ − 2κ)/2.In this range, there does not exist any REE. Although the proof of this pointis long and involved, in the limiting case in which the agent receives no infor-mation at all (i.e., κ → ∞), it is possible to give the following straightforwardand intuitive proof. First, we claim that the any candidate equilibrium in thiscase must have fully revealing prices. To see this, suppose instead that thereis an equilibrium in which two fundamentals θ1 and θ2 = θ1 are associatedwith the same price. Since the agent has no information, his intervention policymust be the same at θ1 and θ2. But then the prices are not equal, giving acontradiction. (It is important to note that both the proposition and this simplelimit argument cover mixed strategies by the agent.) However, there is no fullyrevealing equilibrium either: given the agent’s best response, a fully revealingequilibrium features agent-preferred intervention, a possibility ruled out in theprevious paragraph.

No-equilibrium results may seem difficult to interpret. After all, if takenliterally, a no-equilibrium result implies that the model cannot predict an out-come. Clearly, the fact that our model generates a no-equilibrium result is dueto the REE concept used in the article. In a fully specified trading game, theno-equilibrium outcome can be translated into an equilibrium with a break-down of trade. This is an equilibrium where for some interval of fundamentals,market makers abstain from making markets because they would lose moneyfrom doing so. In Appendix B, we formalize this interpretation by studying theequilibria of a very simple trading game.

2.4.1 Equilibria without agent-preferred intervention. Proposition 2 saysthat the agent-preferred equilibrium is not the only equilibrium when κ isbelow T (θ)/2, but not too low. We next characterize such equilibria. Wedefine an equilibrium as having too much intervention if the agent interveneswith strictly positive probability for some set of fundamentals above θ, andintervenes according to his preferred rule otherwise. Similarly, an equilibriumfeatures too little intervention if the agent intervenes with probability strictlyless than 1 for some set of fundamentals below θ and intervenes according tohis preferred rule otherwise. (Note that in principle, an equilibrium may falloutside both categories, and entail both more intervention than the agent wouldlike at some fundamentals above θ and less intervention than he would like atsome fundamentals below θ. However, we have been unable to establish eitherthe existence or nonexistence of such an equilibrium.)

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X(θ)

X(θ + T(θ))P(θ)

ˆ ˆθ + T(θ) θθθ

Figure 3Security price in an equilibrium with too much intervention

As we will establish, whether equilibria feature too much or too little in-tervention depends on whether the expected security payoff X is concave orconvex. In the case of a convex security, which is our focus, equilibria featuretoo much intervention. Figure 3 depicts an example of such an overinterventionequilibrium.

In the equilibrium depicted in Figure 3, the agent intervenes according to hispreferred rule at fundamentals associated with the left line and the right lineof the pricing function, but intervenes too much at fundamentals associatedwith the middle line. These fundamentals are above θ, yet, in the equilibrium,the agent intervenes with positive probability when they are realized. Thishappens because every fundamental associated with the middle line has a pricethat is identical to that of a fundamental associated with the left line. Sincethe middle line and the left line are close, the agent cannot always tell apartfundamentals associated with these two lines even after observing his owninformation. Since fundamentals associated with the middle line are above θ

and fundamentals associated with the left line are below θ, the agent does notget clear-cut information as to whether to intervene or not. Thus, sometimeswhen the fundamental falls in the middle line, the agent does not have enoughevidence to justify the lack of intervention, and chooses to intervene.

Let us illustrate mathematically what is needed for this equilibrium to hold.Take a pair of fundamentals associated with the left line and the middle lineof Figure 3 that have the same price and call them θ1 and θ2, respectively. Theprobability of intervention at θ1 is 1, and thus the price at θ1 is X (θ1 + T (θ1)).The probability of intervention at θ2 is the probability that the agent observes asignal that is consistent with θ1 conditional on the fundamental being θ2. Giventhe uniform distribution of noise, this probability is equal to 1 − θ2−θ1

2κ. Hence,

the price at θ2 is θ2−θ12κ

X (θ2) + (1 − θ2−θ12κ

)X (θ2 + T (θ2)). For the equilibriumto hold, the prices at θ1 and θ2 have to coincide, and the agent must chooseto intervene when he cannot distinguish between θ1 and θ2. Proposition 3establishes the existence of equilibria of this kind. It also demonstrates that

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when the security is convex, parallel equilibria that exhibit too little interventiondo not exist.

Proposition 3. (i) Suppose that κ < T (θ)/2 is sufficiently close to T (θ)/2.Then, there exist equilibria with too much intervention. In these equilibria, theagent intervenes with positive probability at some fundamentals above θ. Inall other fundamentals, agent-preferred intervention is achieved (that is, thereis intervention with probability 1 below θ and intervention with probability 0above θ).

(ii) Suppose that κ < T (θ)/2. Then, any equilibrium other than the agent-preferred equilibrium entails an intervention probability strictly greater than0 at some fundamental θ > θ.

It is interesting to explore the source of equilibrium multiplicity—i.e., why,when κ is in an intermediate range, both agent-preferred intervention (depictedin Figure 2) and overintervention (depicted in Figure 3) form an equilibrium.Recall that there is an equilibrium with agent-preferred intervention becausewhen intervention is based on the agent’s preferred rule, fundamentals that havethe same price are far enough from each other, and so the signal of the agent,having an intermediate level of precision, is precise enough to enable him totell the fundamentals apart and intervene as he prefers. But, suppose that theagent intervenes with positive probability at some fundamentals that are slightlyabove θ (as in Figure 3). The higher intervention probability increases the priceat these fundamentals and creates a situation where fundamentals that are closerto each other have the same price. This then becomes self-reinforcing and leadsto an equilibrium: as the distance between fundamentals with the same priceshrinks, the agent (with a signal of intermediate precision) cannot always tellthese fundamentals apart, and thus intervenes with positive probability at somefundamentals above θ.

Based on this logic, the result in part (ii) of the proposition seems surprising.After all, it seems straightforward to apply the same logic in the other direc-tion and generate an equilibrium with too little intervention. But, one has toremember that the presence of a force that pushes toward under- or overinter-vention is not enough to guarantee that such an equilibrium will indeed exist.Consider the following intuition for why underintervention is inconsistent witha convex security and moderately informative agent signals. Analogous to theoverintervention case discussed above, in an equilibrium with no interventionabove θ and less than certain intervention below θ, the following equality hasto hold for a continuum of pairs of fundamentals θ1 < θ and θ2 > θ:

X (θ2) =(

1 − θ2 − θ1

2κ

)X (θ1) + θ2 − θ1

2κX (θ1 + T (θ1)). (6)

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When X is convex, this implies that

θ2 >

(1 − θ2 − θ1

2κ

)θ1 + θ2 − θ1

2κ(θ1 + T (θ1)),

or equivalently, T (θ1) < 2κ, which cannot hold when κ < T (θ)/2.

3. Applications

3.1 Corporate governanceThe term corporate governance covers actions taken by various economicagents aiming to control corporate managers and ensure that they are actingin the best interest of shareholders. The idea that market valuations of firms’securities are important for corporate governance has been long recognized.For example, Jensen and Meckling (1979, p. 485) write: “The existence ofa well-organized market in which corporate claims are continuously assessedis perhaps the single most important control mechanism affecting managerialbehavior in modern industrial economies.”

Players in the corporate governance arena include the board of directors,shareholder activists, and others. A large empirical literature shows that theseagents’ actions are correlated with market valuations, and this evidence istypically interpreted as indicating that market valuations affect actions. Oneof the most important decisions that has to be made by the board of directorsis whether to replace an acting CEO. A large literature (e.g., Warner, Watts,and Wruck 1988; Jenter and Kanaan 2006; Kaplan and Minton 2006) on CEOreplacement finds that low market valuations (which presumably indicate poorCEO performance) increase the incidence of CEO replacement.14 Low marketvaluation is also regarded as a key determinant of shareholder activism. Forexample, a large number of the events described by Brav et al. (2008) in theirstudy on hedge-fund activism are triggered by a hedge fund’s belief that thefirm’s market valuation is below its potential value (for a broad literature reviewon shareholder activism, see Gillan and Starks 2007).

Corporate governance actions can be easily mapped into our model. Let θ

denote the expected cash flow of the firm absent intervention by the boardof directors or by the activist, and let T (θ) denote the change in expectedcash flow as a result of taking the action. Let C denote the private cost thatdirectors or activists have to bear when intervening. These costs can be quitesignificant. In the context of the board of directors replacing the CEO, C canrepresent a reputational cost or a loss of private benefit resulting from fightingagainst an acting CEO. Taylor (2008) estimates the private cost borne bydirectors to be 5.6% of the firm value, on average. In the context of shareholder

14 The reliance of directors on market prices has presumably increased over time as more directors are nowindependent of the firm and hence have little direct information on its operations (see Gordon 2007).

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activism, we are not aware of any formal estimate of the private costs borneby activists, but it is widely agreed that shareholders wishing to intervene inthe firm’s business have to incur significant costs to cover legal battles andconvince other shareholders to vote for their proposal (see, e.g., Gillan andStarks 2007). Likewise, in this context it seems reasonable to suppose that theagent’s benefits from intervention, V , are decreasing in the fundamental. Underthe additional mild assumption that the agent benefits from intervention only ifthe expected cash flow is increased, then intervention is a corrective action forall fundamentals θ ≤ θ.

3.2 Bank supervisionIn the United States, a bank regulator/supervisor who believes that a bankis performing poorly possesses a variety of mechanisms by which he canattempt to improve the bank’s health. These range from encouraging bankmanagement to correct identified problems to formal agreements that restrictcapital distributions and management fees, limit bank activities, or even dismisssenior officers or directors. Under some circumstances, these regulatory actionsare even mandated by the prompt corrective action provisions in the FederalDeposit Insurance Corporation Improvement Act of 1991.15 Furthermore, asrecent events have demonstrated, regulators can provide liquidity to a bank thatis having trouble borrowing in the interbank market and can offer guaranteesfor some of the bank’s bad assets.

As Feldman and Schmidt (2003) and Burton and Seale (2005) document,bank supervisors in the United States make substantial use of market infor-mation in assessing a bank’s condition. Moreover, many proposals call forstrengthening the reliance on market data. For example, a recent proposalsuggests requiring banks to regularly issue subordinated debt, partly so thatsupervisors can use the price of debt to monitor the health of issuing banks(see Evanoff and Wall 2004; Herring 2004). This proposal is based in parton evidence that bank security prices reflect underlying risk and contain in-formation that regulators do not have—see, for example, Krainer and Lopez(2004) and the surveys by Flannery (1998) and Furlong and Williams (2006).In a similar fashion, Gary Stern, the president of the Federal Reserve Bankof Minneapolis, argues that market data complement supervisory assessmentsbecause they are generated “on a nearly continuous basis” by “a very large

15 As an example of the type of actions that U.S. regulators may take, consider the following 2002 written agreementwith PNC Bank, which was instigated by accounting irregularities. To ensure that PNC implemented amongother things the necessary risk management systems and internal controls, the bank was required to hire anindependent consultant to “review the structure, functions, and performance of PNC’s management and theboard of directors oversight of management activities. . . . The primary purpose of the [review] shall be to assistthe board of directors in the development of a management structure that is adequately staffed by qualifiedand trained personnel suitable to PNC’s needs.” (Board of Governors of the Federal Reserve System, DocketNo. 02-011-WA/RB-HC. Written Agreement by and between PNC Financial Services Group, Inc., Pittsburgh,PA, and the Federal Reserve Bank of Cleveland, July 2, 2002.)

For more details on actions that U.S. regulators can take, see Spong (2000). Appropriate regulation is thesubject of a substantial literature; see, e.g., Morrison and White (2005) for one positive theory of bank regulationalong with the references cited therein.

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number of participants [who] have their funds at risk of loss” and are “nearlyfree to supervisors.”16

The mapping to our model is again straightforward. One simple interpretationof our model in the context of bank supervision is that the supervisor is interestedin maximizing total surplus. By intervening in the bank’s business, he canincrease the expected cash flows by T (θ) (which coincides here with V (θ)), buthe also has to bear a private cost of C . Hence, the supervisor wishes to interveneif and only if T (θ) is greater than C . Another way to think about the supervisor’sproblem is that he is interested in protecting depositors and thus will interveneonly when the probability that the bank will not have enough resources topay depositors is high. In this case, V (θ) is clearly different from T (θ): V (θ)represents the benefit to the deposit insurer from intervention,17 while T (θ) isthe change in total expected cash flow as a result of intervention.18

A key element of our analysis is that the security price is nonmonotonicwith respect to the fundamental due to potential intervention. In the contextof bank supervision, there is empirical evidence for such nonmonotonicity.DeYoung et al. (2001) show that the price of bank debt increases in responseto an unexpectedly poor exam rating for lower quality banks. Related, Covitz,Hancock, and Kwast (2004) and Gropp, Vesala, and Vulpes (2006) documentthat only a weak relation between the market price of debt and risk is observedwhen the government support of debt holders is more likely.

3.3 Managerial investment decisionsA growing empirical literature demonstrates that firm managers use informa-tion from the market price of their firms’ securities when making corporateinvestment decisions (see Luo 2005; Chen, Goldstein, and Jiang 2007; Bakkeand Whited, forthcoming). To fix ideas, consider an acquisition decision. Aftera firm announces that it is going to acquire another firm, its stock price will reactto reflect the beliefs in the market about whether the acquisition is a good ideaor not. Luo (2005) provides evidence consistent with the idea that managers

16 See http://www.minneapolisfed.org/pubs/region/01-09/stern.cfm.

17 Note that although the expected payout of a deposit insurer decreases in θ, the reduction in the payout associatedwith intervention does not necessarily decrease. However, one can show that under very mild assumptionsV either decreases, or increases and then decreases. Consequently, limiting attention to a range of relevantfundamentals [θ, θ] and assuming that V (θ) > C > V (θ), there exists a unique θ such that V (θ) > C if and onlyif θ < θ. This is the only property of V that we use in our analysis. Details are contained in an earlier draft andare available upon request.

18 In the world of regulation and policy making, learning from market prices occurs also outside the context of banksupervision. Piazzesi (2005) demonstrates the importance of accounting for the dual relation between monetarypolicy and market prices in explaining bond yields. Another example is the Sarbanes-Oxley Act of 2002. Section408 of the act calls for the Securities and Exchange Commission to consider market data—namely, share pricevolatility and price-to-earnings ratios—when deciding whether to review the legality of a firm’s disclosures. Afinal example is class action securities litigation. Courts in the United States use share price changes as a guidefor determining damages (see, e.g., Cooper Alexander 1994).

Other theoretical papers study different dimensions of market-based regulation. Faure-Grimaud (2002); Rochet(2004); and Lehar, Seppi, and Strobl (2007) study the effect of market prices on a regulator’s commitment ability.Morris and Shin (2005) argue that transparency by the central bank may be detrimental, as it reduces the abilityof the central bank to learn from the market.

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use the information in the reaction of the market to decide whether to cancelthe acquisition.

In the language of our model, θ is the expected cash flow of the (potentially)acquiring firm, assuming that the acquisition goes through. The manager isthe agent who can intervene and cancel the acquisition. If the acquisition iscancelled, the cash flow of the firm will be θ + T (θ). However, if the managercancels the acquisition, he will have to bear a private cost C . This cost couldrepresent a forgone private benefit of control that the manager could achieve ifthe acquisition took place, or a reputational cost that the manager bears if theacquisition is cancelled. Under the mild assumption that the manager benefitsonly from cancelling cash-flow destroying acquisitions, the intervention is acorrective action.19

4. Robustness and extensions

We now turn to discuss some robustness issues and extensions of the basicmodel.

4.1 Shapes of the security and the intervention function TIn Section 2.4, we presented the equilibrium analysis in the case of a non-monotone price function under the assumptions that X (θ) and X (θ + T (θ)) areconvex with respect to θ, and that |T ′(θ)| is sufficiently small in the interval be-tween θ − 2κ and θ + 2κ. The latter assumption was used in our proofs to implythat U (θ) increases in θ. We now briefly discuss the results under alternativeassumptions. Full details are available upon request.

Maintaining the assumption of convexity, but assuming that |T ′(θ)| is large,and hence U (θ) decreases, we can again establish the uniqueness of the equi-librium with agent-preferred intervention when κ is below some threshold, andthe nonexistence of equilibrium when κ is large. The only difference relativeto the results presented in the previous section is that under this alternative as-sumption, we cannot find an equilibrium without agent-preferred interventionfor an intermediate range of κ.20

19 The empirical analysis of Kau, Linck, and Rubin (2008) is consistent with managers considering both the marketreaction to the acquisition announcement and their own private benefits from the acquisition when decidingwhether or not to go ahead with the acquisition.

20 A natural question is how small |T ′(θ)| has to be in order for equilibria without agent-preferred intervention toarise with a convex security. Inspecting the proof of Proposition 3, one can see that two conditions are required.First, and as noted, U must be increasing over the range of fundamentals where intervention probabilities otherthan 0 or 1 are possible—i.e., between θ − 2κ and θ + 2κ (see Lemma 1 of Appendix A). By straightforwarddifferentiation, U is increasing if and only if |T ′(θ)|X ′(θ + T (θ)) < X ′(θ + T (θ)) − X ′(θ). Second, |T ′(θ)| must

be small enough that |T ′(θ)|X ′(θ + T (θ)) < X ′(θ + T (θ)) − U (θ)2κ

(see the proof of Proposition 3).Numerical simulations suggest that these conditions hold for a reasonably large range of parameters. One

example in which both conditions are satisfied is as follows. The security is equity; the firm’s total debt isD = 1.5; the cutoff θ = 1; the cash-flow shock δ is normally distributed, with a standard deviation of 0.5; theeffect T of intervention is linear, with T (θ) = 0.25 and T ′ = −0.2T (θ); and the precision of the agent’s signalis determined by κ = 0.4T (θ).

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We now move to a concave security, which is most relevant for the applicationof bank supervision where regulators learn from the price of debt.21 We againfind that the equilibrium with agent-preferred intervention is unique whenκ is below some threshold, and that no equilibrium exists when κ is large.The difference now is in what kind of equilibria arise without agent-preferredintervention. While for a convex security we establish that for an intermediaterange of κ, there exist equilibria with too much intervention, and there are noequilibria with too little intervention, the opposite holds for a concave security.That is, if the security is concave with respect to the fundamental, there exists anintermediate range of κ for which there are equilibria with too little intervention,but no equilibria with too much intervention.

To summarize, a general result for various assumptions about the parametersis that agent-preferred intervention is obtained as a unique equilibrium whenthe information gap between the market and the agent is small (i.e., when κ

is small), while no equilibrium exists—or a market breakdown occurs—whenit is large. Different equilibria without agent-preferred intervention may existwhen κ is in an intermediate range, depending on the curvature of the securityand the sensitivity of the effect of intervention to the fundamental.

4.2 Information structureThus far we assumed that the agent has strictly less information than the market,since the market observes a state variable θ, while the agent observes only anoisy signal of θ: φ = θ + ξ. This information structure is restrictive becausein the applications we consider, agents—e.g., directors, activists, regulators,and managers—may have access to some information that is not available tothe market.

To explore the robustness of our analysis to this assumption, we considerthe following set of alternative assumptions, which allow for the possibilitythat the agent’s information is superior to the information observed by marketparticipants. Suppose that the agent would like to intervene if and only ifan underlying state variable, ψ, is below some critical level, ψ. The marketobserves a signal θ, which is an unbiased forecast of ψ (θ = ψ + ζ). The agentsometimes has better information than the market and sometimes has worseinformation than the market. In particular, suppose that with probability μ theagent observes ψ, while with probability 1 − μ, he observes φ = θ + ξ (as inour model). The agent knows the accuracy of the information he acquires—i.e.,

21 In fact, thinking about the typical financial structure of banks, debt securities are usually convex for lowfundamentals and concave for high fundamentals. Economically, a convex-then-concave shape arises becausedebt is junior to deposits but senior to equity claims. Hence, when the fundamentals are low, debt holders arelikely to be the residual claimants, which leads to a convex shape, while for high fundamentals they are likely tobe paid in full, which leads to a concave shape.

In the article, we characterize equilibrium outcomes for either a convex (our main focus in Section 2) or aconcave (briefly described here) X (·) function. Hence, for the case of debt, we essentially characterize resultsfor situations where the relevant fundamentals (i.e., some range around θ) are either in the range where debt isconcave or in the range where debt is convex. In addition, most of our results also hold for a convex-then-concavesecurity. Details are available from us upon request.

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whether he observes ψ or φ. The market does not observe what informationthe agent acquires. For μ sufficiently small, the previous analysis goes throughcompletely under this richer set of assumptions, as follows.

In the extended model, if the agent observes ψ (with probability μ), heignores the market price and chooses to intervene if and only if ψ is belowψ. If he observes φ (with probability 1 − μ), he acts as in our basic modeland chooses to intervene if and only if E[θ|P,φ] is below some θ. The markettakes these different scenarios into account when pricing the firm’s security.Specifically, let X∗(θ) denote the expected value of the security given marketsignal θ and given that the agent sees ψ and intervenes according to his preferredrule. Then, carrying the logic in Equation (3) to the extended model, the priceof the security in the market is

P(θ) = μX∗(θ) + (1 − μ)[X (θ) + Eφ[I (P(θ),φ)|θ]U (θ)],

where I (P,φ) denotes the agent’s intervention decision when he does notobserve ψ, but instead sees just the security price P and his noisy signal, φ.Defining X and U by

X (θ) = μX∗(θ) + (1 − μ)X (θ)

U (θ) = μX∗(θ) + (1 − μ)X (θ + T (θ)) − X (θ) = (1 − μ)U (θ),

the pricing equation can be rewritten in a way that makes it analogous toEquation (3) in our basic model:

P(θ) = X (θ) + Eφ[I (P(θ),φ)|θ]U (θ).

The extended model is thus analogous to our basic model with the functionsX (θ) and U (θ) replacing X (θ) and U (θ), respectively, and with T replaced byT defined by

X (θ + T (θ)) = μX∗(θ) + (1 − μ)X (θ + T (θ)).

Our analysis of the basic model uses the following key properties: X andX (· + T (·)) are increasing; X and X (· + T (·)) are either convex or concave;and T is decreasing. All these properties are inherited by X , X (· + T (·)), andT when μ is sufficiently small. Moreover, T (θ) > 0 whenever T (θ) > 0 and μ

is sufficiently small. Then, the analysis of our model goes through completelyunder the richer set of assumptions.

In summary, the setting analyzed in this section serves to demonstrate thatthe assumptions of our main model are not that restrictive, and that our analysisextends to a setting where the agent sometimes has better information than themarket. An alternative setting to consider would be one where the market andthe agent are treated more symmetrically. That is, suppose again that the agentcares about the state variable ψ, but that both the market and the agent observenoisy signals of ψ: the market observes θ = ψ + ζ and the agent observes

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φ = ψ + ξ. Unfortunately, this framework loses tractability very fast, and itdoes not enable us to analytically conduct most of the analysis conducted in thearticle. We are able to confirm only a pair of basic results with this alternativeframework. First, if the agent has no signal, no equilibrium exists.22 Second,if both the market’s and agent’s signals are relatively precise, an equilibriumexists and converges to an agent-preferred equilibrium as the market’s signalbecomes infinitely precise. In this sense, the agent-preferred equilibrium of ourbasic model is robust. Details are available from us upon request.

4.3 State variables other than expected cash flowIn our basic model, the market observes expected cash flow θ, and interventionaffects expected cash flow. However, security values also depend on highermoments of the distribution of cash flow, and it is possible that the agent wantsto learn the value of some higher moment rather than the expected cash flow.For example, a bank regulator may care about the variance of bank cash flows,and intervention may be aimed at preventing excessive risk taking.

Provided that the information asymmetry between the market and agent issummarized by a one dimensional state variable, our analysis extends to suchsettings, given parallel assumptions to those we make in our model. That is, forany underlying state variable θ (e.g., the inverse of the variance of cash flows),one can define X (θ) and X I (θ) as the expected security values without and withintervention, and T (θ) by X (θ + T (θ)) = X I (θ). Then, provided T is weaklydecreasing, and X and X I are increasing and are either both convex or bothconcave, our analysis applies.

5. Making learning more efficient

Returning to our main model, we now investigate ways to overcome the probleminvolved in market-based corrective actions. First and foremost, it should benoted that a main insight of our model is the strong complementarity betweenthe market’s information and the agent’s information. To be able to implementa successful market-based intervention policy, the agent still needs to producea reasonably precise signal of his own. Thus, learning from the market cannotperfectly substitute for direct sources of information. This is perhaps the mainnormative implication of our model, and is obtained despite the fact that ourmodel endows the market with perfect information about the fundamentals.The role of information in our model is to help the agent tell the extent to whichthe market price reflects information about the fundamental and the extent towhich it reflects information about the expected agent’s action. In that sense,the private information in our model plays a somewhat unusual role.

We next study whether there are alternatives to the agent generating a precisesignal for which market-based intervention will work. The first alternative we

22 At this extreme, the model under discussion coincides with our main model.

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consider is for the agent to learn from the prices of multiple securities. Thesecond alternative is to improve transparency by disclosing the agent’s signalto the market. The third alternative is to issue a security that directly predictswhether the agent is going to intervene. We show that each one of these measuresameliorates the agent’s inference problems—although as we describe below,nontrivial conditions must be met for each measure to be feasible in the firstplace. Finally, we consider the possibility that the agent can commit ex anteto an intervention rule based on the realized price. We show that this does notresolve the agent’s inference problems.

5.1 Multiple securitiesThus far we have restricted attention to the case in which the agent observesonly one price, that of a convex security. As noted, parallel results hold forthe case in which the agent observes the price of a concave security insteadof that of a convex security. The only difference between the two cases isthat in the range of multiple equilibria, underintervention is possible with aconcave security, while overintervention is possible with a convex security. Akey question is whether it helps if both these securities trade publicly, and theagent learns from the prices of both.

It turns out that observing the prices of both securities resolves the problemof multiple equilibria when the agent’s signal is moderately precise, but doesnot solve the problem of no REE when the agent’s signal is imprecise.

Proposition 4. (i) Suppose that κ < T (θ)/2 and that the agent observes theprice of both a strictly concave and a strictly convex security. Then the agent-preferred equilibrium is the unique equilibrium.

(ii) Suppose that κ > T (θ − 2κ)/2 and that the agent observes the price ofboth a concave and a convex security. Then no equilibrium exists.

To gain intuition for the first part, recall the results of the previous sections.There, we showed that when the agent’s information is moderately precise,there may exist equilibria with too much or too little intervention, in additionto the equilibrium with agent-preferred intervention. We also showed that anequilibrium with too much intervention requires that the security whose pricethe agent observes be convex, while an equilibrium with too little interventionrequires that the security be concave. Thus, in this range, observing both theprice of a concave security and the price of a convex security eliminates theequilibria without agent-preferred intervention.

This result suggests that there is a significant benefit to learning from twodifferent securities. Thus, for example, bank regulators can be instructed tolearn simultaneously from the prices of bank debt and equity, instead of justfrom the price of bank debt. It is important to note that this implication of themodel requires that two distinct securities trade in well-functioning markets.This condition is not always satisfied.

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In addition, even if this condition is met, the agent still faces inferenceproblems when his information is imprecise. Specifically, as the second partof the proposition shows, even multiple security prices do not help the agentwhen κ > T (θ − 2κ)/2. The basic intuition utilizing the limiting case in whichκ → ∞ is the same as that provided for the nonexistence result in the case ofone security.

5.2 Transparency/disclosureWe now return to the case of one traded convex security and assume that theagent makes public his own signal φ before the market price is formed. Inmost corporate contexts, this would be termed “voluntary disclosure,” whilein the bank regulation context one might speak of regulatory “transparency.”Our analysis implies that this form of transparency improves the agent’s abilityto make use of market information. Specifically, transparency resolves theproblem of multiple equilibria when the agent’s signal is moderately precise,but it does not solve the problem of no REE when the agent’s signal is imprecise.The argument is as follows.

Under the “transparency” regime in which the agent truthfully announceshis signal φ, the equilibrium pricing function depends on both the fundamentalθ and the agent’s signal φ. Consider a specific realization φ∗ of the agent’ssignal, along with any pair of fundamentals θ1 and θ2 such that φ∗ is possibleafter both. The prices at (θ1,φ

∗) and (θ2,φ∗) must differ. If, instead, the prices

coincided, the intervention decisions would also coincide, but in this casethe prices would not be equal after all. It follows that all fundamentals θ

for which the agent’s signal φ∗ is possible must have different prices givenrealization φ∗—that is, given φ∗ prices are fully revealing. This argumenttogether with the fact that the agent always chooses his best response impliesthat the only candidate equilibrium features agent-preferred intervention. Assuch, transparency eliminates the equilibria of Proposition 3. The intuition isthat these equilibria were based on the market not knowing the agent’s action,a problem that is solved once the agent discloses his signal truthfully.

Now, when κ < T (θ)/2, agent-preferred intervention is indeed an equilib-rium, with prices P(θ,φ) = X (θ) + U (θ) for θ ≤ θ and P(θ,φ) = X (θ) forθ > θ. In contrast, when κ > T (θ)/2, agent-preferred intervention is not anequilibrium. To see this, if we suppose to the contrary that it were an equilib-rium, then there exist fundamentals θ1 and θ2 and an agent’s signal realizationφ ∈ [θ1 − κ, θ1 + κ] ∩ [θ2 − κ, θ2 + κ] such that (θ1,φ) and (θ2,φ) have thesame price, in contradiction to the above. It follows that for κ > T (θ)/2, thereis no equilibrium.

Although a policy of transparency improves the agent’s ability to infer funda-mentals from market prices, in practice there may be limits to its viability. Forexample, take the case of bank supervision: if a bank knows that the regulatorwill make its information public, it may be less inclined to grant easy accessto the regulator in the first place. In this sense, it is possible that transparency

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would serve to increase κ, potentially making the regulator’s inference problemworse instead of better.

5.3 Prediction marketsNeither of the measures discussed so far allows the agent to infer the fundamen-tal when his own information is poor (κ > T (θ − 2κ)/2). The next possibilitywe discuss is the creation of a “prediction market” in which market participantstrade a security that pays 1 if the agent intervenes, and 0 otherwise. Clearly,such a market is feasible only if the agent’s intervention is publicly observableand verifiable23—a condition that is not required in any of our analysis to thispoint, and in practice may fail to hold (for example, verifying the actions ofshareholder activists is quite difficult). However, if such a market could becreated, its existence would render agent-preferred intervention as the uniqueequilibrium irrespective of the quality of the agent’s information.

More formally, suppose that in addition to a standard security market, aprediction market of the type described is feasible and exists. Let Q be theprice of the security in the prediction market, with P being the price of theequity security as before. The agent’s intervention policy I can now dependon Q in addition to P and his own signal φ. The REE pricing condition forthe prediction-market security is Q(θ) = Eφ[I (P(θ), Q(θ),φ)|θ]. Under theseconditions we obtain the following:

Proposition 5. If the market trades both a standard equity security and theprediction-market security, then for all κ the unique equilibrium of the economyfeatures agent-preferred intervention.

The intuition behind this result is the following: a regular equity securitymay have the same price for different fundamentals because the probability ofintervention is different across these fundamentals. But, once the prediction-market security is traded, the probability of intervention can be inferred fromits price, and thus the fundamental can be inferred from the combination ofits price and the price of equity. This implies that the agent will interveneaccording to his preferred rule in equilibrium.

5.4 CommitmentThus far we have assumed that the agent acts in an ex post optimal waygiven the price and his signal. A natural question is whether the agent canachieve his preferred intervention by committing ex ante to an interventionrule as a function of the realized price. To answer this question, we assumethat the agent can commit ex ante to an intervention policy that is a functionof the price only. This last assumption is natural given that committing to anintervention rule that is based on the publicly observed price may be feasible,while committing to an intervention rule that is based on a privately observed

23 For monetary policy actions, the Fed Funds futures market serves as just such a market.

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signal is probably not. In view of the agent’s commitment, for this subsectiononly we drop the requirement that the best-response condition (4) must besatisfied in equilibrium.

The main thing to note about this case is that an equilibrium under com-mitment must entail fully revealing prices—i.e., in such an equilibrium everyfundamental must be associated with a different price. This is because theagent’s intervention decision is now based only on the price. As a result, if twofundamentals had the same price, they would also have the same probability ofintervention, which would generate different prices. Thus, finding the optimalcommitment policy for the agent boils down to finding the price function thatmaximizes the agent’s ex ante value function, subject to the constraint that theprice function fully reveals the fundamentals.

The fact that the price function must be fully revealing implies that theagent cannot achieve his preferred intervention under commitment. This isbecause, as we saw in Figure 2, agent-preferred intervention generates a pricefunction that is not fully revealing—it has different fundamentals associatedwith the same price. The following proposition establishes a stronger result onthe effectiveness of commitment. It says that, under commitment, the agentwill end up deviating from his preferred intervention policy over a set offundamentals that is at least of size T (θ).

Proposition 6. If the agent commits ex ante to an intervention policy basedon the realization of the price of one security, he will not be able to achievehis preferred intervention. The set of fundamentals at which the agent deviatesfrom his preferred intervention policy is at least of size T (θ).

Proposition 6 says that commitment by the agent does not allow him to fullylearn from the price and then use that information as he would like. However,when the agent’s information is poor (κ large), commitment does at least ensurethat an equilibrium exists, and so (in our interpretation) avoids the problemsassociated with market breakdown. It is important to note that the welfarelosses associated with commitment and no-commitment are hard to compare.The reason is that in both cases, the agent’s action partially reflects the market’sinformation θ, but the distance between the agent’s equilibrium and preferredactions differs across the two cases. Moreover, the cost of the agent deviatingfrom his preferred action is in turn hard to compare with the direct welfare costof a breakdown of trade in some fundamentals that arises in the no-commitmentcase.

6. Conclusion

We study a rational expectations model of market-based corrective actions.A key issue is that prices reflect both firm fundamentals and expectations ofcorrective actions. In a wide range of cases, this generates nonmonotonicity ofthe price with respect to fundamentals. When this happens, the agent taking

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the decision on the corrective action cannot easily extract information from theprice to make an efficient intervention decision. We provide a characterizationof the equilibrium outcomes of our model and show that the ability of the agentto extract information from the market depends on the gap between his and themarket’s information quality. We also relate equilibrium outcomes to the typeof security whose price the agent observes. Convex securities may lead to toomuch intervention, while concave securities may lead to too little.

A key normative implication of our analysis is that market data and privateinformation should be treated as complements, in the sense that the agent’s owninformation is crucial for him in understanding whether shifts in market pricesare due to changes in fundamentals or to changes in expectations regardinghis own actions. We also provide implications for the potential efficacy of anumber of measures intended to improve learning from prices. Finally, wederive positive empirical implications on the relation between market pricesand corrective actions that are based on them.

The general insights from our analysis can be applied to many settings inwhich individuals use information from market prices to take actions that havea corrective effect on the value of the security. Examples include the decision ofthe board of directors on whether to replace a CEO, the decision of shareholderactivists on whether to take actions to intervene in the operations of the firm,the decision of bank supervisors on whether to take actions to improve thehealth of a financial institution, and the decision of a firm manager on whetherto cancel a previously announced acquisition.

These applications have been the subject of many empirical papers. Ourmodel has strong implications on how to conduct empirical analysis in theseand other related settings. In particular, two key features of the model have to betaken into account. First, if agents (i.e., regulators, directors, activists) use themarket price in their intervention decision, there will be dual causality betweenmarket prices and the intervention decision: market prices will reflect the agent’saction and affect it at the same time. In the context of shareholder activism inclosed-end funds, Bradley et al. (forthcoming) conduct empirical analysis thattakes into account this dual causality. Second, when the information that agentshave outside the financial market is not precise enough, our model generatesequilibrium indeterminacy, which might make the relation between marketprices and intervention more difficult to detect.

Appendix A: Proofs

The following straightforward result is used in several places:

Lemma 1. Suppose that T (θ) > 0, and define θ < θ by θ + T (θ) = θ. In any equilibrium,

Pr(I |θ) ={

1 if θ < max{θ − 2κ, θ − T (θ)}0 if θ > min{θ + 2κ, θ + T (θ)} .

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Proof of Lemma 1. Consider a fundamental θ < θ − 2κ. At this fundamental, the agent observesonly signals below θ − κ. Such signals are never observed after any fundamental θ ≥ θ. As such,when the fundamental is θ the agent knows that the fundamental lies to the left of θ and interveneswith probability 1. By a similar argument, the agent never intervenes if θ > θ + 2κ.

Next, consider a fundamental θ < θ − T (θ) = θ. In any equilibrium, the price at θ is boundedabove by X (θ) + U (θ) = X (θ + T (θ)) < X (θ + T (θ)) = X (θ). Moreover, any fundamental θ ≥ θ

has a price that satisfies P(θ) ≥ X (θ) ≥ X (θ). Thus, in any equilibrium, if θ < θ − T (θ), thenθ cannot share a price with any fundamental above θ. Again, the agent intervenes withprobability 1.

Finally, consider a fundamental θ > θ + T (θ). In any equilibrium, the price at θ strictly exceedsX (θ + T (θ)). Moreover, any fundamental θ ≤ θ has a price that satisfies P(θ) ≤ X (θ + T (θ)) ≤X (θ + T (θ)). Thus, in any equilibrium, if θ > θ + T (θ), then θ cannot share a price with anyfundamental below θ. Again, the agent intervenes with probability 0.

Proof of Proposition 1. Existence is immediate. For uniqueness, suppose to the contrary that anequilibrium without agent-preferred intervention exists. Any such equilibrium must feature a priceP shared by a set of fundamentals �P , where �P has at least one element strictly less than θ andat least one element strictly greater than θ. Fix any fundamental θ2 ∈ �P that strictly exceeds θ.Let q(θ) denote the intervention probability at fundamental θ. Since all fundamentals in �P sharethe same price, the following must hold for every θ ∈ �P :

q(θ2)X (θ2 + T (θ2)) + (1 − q(θ2))X (θ2) − q(θ)X (θ + T (θ)) − (1 − q(θ))X (θ) = 0. (7)

The left-hand side of Equation (7) can be rewritten as

q(θ2)(X (θ2 + T (θ2)) − X (θ + T (θ)))

+ (1 − q(θ2))X (θ2) − (q(θ) − q(θ2))X (θ + T (θ)) − (1 − q(θ))X (θ). (8)

Note that for any θ ∈ �P that is below θ, the facts that X (θ + T (θ)) is increasing and T (θ) isnegative imply X (θ2) > X (θ) ≥ max{X (θ + T (θ)), X (θ)}. If q(θ2) = 0, this delivers an immedi-ate contradiction since q(θ) − q(θ2) ≥ 0 and so, (1 − q(θ2))X (θ2) > (q(θ) − q(θ2))X (θ + T (θ)) +(1 − q(θ))X (θ).

The remainder of the proof deals with the case in which q(θ2) > 0. Define θ∗ = sup �P ∩ [θ, θ]and observe that for any θ ∈ �P ∩ [θ, θ],

q(θ) − q(θ2) = 1

2κ

(∫ θ+κ

θ−κ

I (P,φ) dφ −∫ θ2+κ

θ2−κ

I (P,φ)dφ

)

= 1

2κ

(∫ θ2−κ

θ−κ

I (P,φ) dφ −∫ θ∗+κ

θ+κ

I (P, φ) dφ −∫ θ2+κ

θ∗+κ

I (P,φ) dφ

)

= 1

2κ

(∫ θ2−κ

θ−κ

I (P,φ) dφ −∫ θ∗+κ

θ+κ

I (P, φ) dφ

),

where the final equality follows since the price P and a signal above θ∗ + κ together tell the agentthat the fundamental definitely exceeds θ. It follows that for any ε > 0, there exists some θ ∈�P ∩ [θ, θ] such that q(θ) − q(θ2) > −ε. Hence for any ε′ > 0, there exists some θ ∈ �P ∩ [θ, θ]such that

(1 − q(θ2))X (θ2) − (q(θ) − q(θ2))X (θ + T (θ)) − (1 − q(θ))X (θ) > −ε′.

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Finally, since q(θ2)(X (θ2 + T (θ2)) − X (θ + T (θ))) > 0 for θ ≤ θ, it is possible to choose θ ∈�P ∩ [θ, θ] such that Equation (8) is strictly positive, contradicting Equation (7) and completingthe proof.

Proof of Proposition 2 (existence of equilibrium with agent-preferred intervention). The maintext shows both that there is an equilibrium with agent-preferred intervention if κ < T (θ)/2, andthat there is no such equilibrium if κ > T (θ)/2.

Proof of Proposition 2 (equilibria without agent-preferred equilibria). The proof is by con-struction, and covered by Proposition 3.

Proof of Proposition 2 (uniqueness of agent-preferred intervention for κ small enough). For thispart of the proof, we need to be more mathematically precise in our treatment of probabilities andexpectations than is the case elsewhere in the article. In particular, unlike elsewhere in the article,we must assign conditional expectations and probabilities in cases where the conditioning set hasinfinitely many members yet is still null. Formally, consider the probability space ([θ, θ],B,μ),where B is the Borel algebra of [θ, θ], and where the fundamental θ is distributed according to theprobability measure μ.

Let κ ∈ (0, T (θ)/2) be such that U (θ)2κ

− (1 + T ′(θ))X ′(θ + T (θ)) > 0 for all θ ∈ [θ, θ + T (θ)],and fix an arbitrary κ ∈ [0, κ]. We show, by contradiction, that in any equilibrium, agent-preferredintervention occurs almost surely. Suppose to the contrary that there exists an equilibrium inwhich the intervention decision differs from agent-preferred intervention over a nonnull set offundamentals.

Throughout the proof we use the following definitions. Let P be the set of nonrevealing prices.For each nonrevealing price P ∈ P, let �P be the set of fundamentals associated with that price.Let � = ⋃

P∈P �P be the set of all fundamentals with a nonrevealing price. By hypothesis, � hasa strictly positive measure.

Claim A. � ∩ [θ, θ + 2κ] has a strictly positive measure.

Proof of Claim A. Consider the conditional probability Pr(�P ∩ [θ, θ + 2κ]|�P ). Clearly, itequals Pr(� ∩ [θ, θ + 2κ]|�P ). Moreover,

∫θ∈�

Pr(� ∩ [θ, θ + 2κ]|�P(θ))μ(dθ) = Pr(� ∩ [θ, θ + 2κ]|�).

Suppose that contrary to the claim, � ∩ [θ, θ + 2κ] is null. In this case, Pr(� ∩ [θ, θ + 2κ]|�P(θ)) =0 for almost all θ in �. But then the agent would intervene according to his preferred rule for almostall θ ∈ �: he would intervene with probability 1 at almost all θ ∈ �, since almost all members of� lie below θ. Since intervention that is not according to the agent’s preferred rule can potentiallyhappen only at θ ∈ �, this contradicts an equilibrium in which the agent intervenes not accordingto his preferred rule over a nonnull set of fundamentals, and completes the proof of Claim A.

For any signal realization φ, the agent knows that the true fundamental lies in the interval[φ − κ, φ + κ]. As such, for a price P ∈ P and signal φ, the agent’s expected payoff (net of costs)from intervention is

v(P, φ) ≡ Eθ[V (θ) − C |θ ∈ �P ∩ [φ − κ, φ + κ]]. (9)

The heart of the proof lies in establishing the following:Claim B. For any P ∈ P , (1) sup �P ∩ [θ − 2κ, θ] = θ and (2) v(P, φ = θ + κ) ≥ 0 for anyθ ∈ �P ∩ [θ − 2κ, θ].

Proof of Claim B. Let θ1 and θ2 ∈ (θ1, θ1 + 2κ] be an arbitrary pair of members of �P suchthat θ1 ≤ θ and θ2 ≥ θ (clearly all members of �P cannot lie to the same side of θ, and at least

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one such pair must lie within 2κ of each other). Since θ1 and θ2 have the same price:

X (θ1) + U (θ1)

2κ

∫ θ1+κ

θ1−κ

I (P,φ) dφ = X (θ2) + U (θ2)

2κ

∫ θ2+κ

θ2−κ

I (P, φ) dφ.

When |T ′(θ)| is sufficiently small, U is increasing, and so U (θ2) > U (θ1). It follows that

X (θ1) + U (θ1) ≤ X (θ2) + U (θ2)

2κ

(∫ θ2+κ

θ2−κ

I (P,φ) dφ +∫ θ1+κ

θ1−κ

(1 − I (P, φ)) dφ

).

Equivalently,

X (θ1 + T (θ1)) ≤ X (θ2 + U (θ2)

2κ

(θ1 − θ2 + 2κ+

∫ θ2−κ

θ1−κ

(1 − I (P,φ)) dφ +∫ θ2+κ

θ1+κ

I (P,φ) dφ

).

(10)

Define θ∗1 = sup �P ∩ [θ − 2κ, θ] and θ∗

2 = inf �P ∩ [θ, θ + 2κ].Suppose that either v(P, φ = θ1 + κ) < 0 or θ∗

1 < θ. In the former case, v(P, φ) < 0 for anysignal φ above θ1 + κ (since if v(P, φ) is strictly negative for some φ, the same is true for all higherφ). In the latter case, any signal φ above θ∗

1 + κ rules out that θ ≤ θ. As such, I (P,φ) = 0 for allφ > θ1 + κ in the former case, and φ > θ∗

1 + κ in the latter case. Since both sides of Equation (10)are continuous in θ1 and θ2, it follows that

X (θ + T (θ)) ≤ X (θ∗2) + U (θ∗

2)

2κ

(θ − θ∗

2 + 2κ +∫ θ∗

2−κ

θ−κ

(1 − I (P,φ)) dφ

)

for θ = θ1 in the former case, and θ = θ∗1 in the latter case. Certainly I (P, φ) = 1 for all φ < θ∗

2 − κ,since for these signal values the agent knows that the fundamental lies to the left of θ. Thus thefunction Z defined by

Z (θ, θ2) ≡ X (θ2) + U (θ2)

2κ(θ − θ2 + 2κ) − X (θ + T (θ))

is weakly positive at (θ, θ2) = (θ1, θ∗2) in the former case, and at (θ∗

1, θ∗2) in the latter case. However,

Z (θ∗2, θ

∗2) = X (θ∗

2) + U (θ∗2) − X (θ∗

2 + T (θ∗2)) = 0

Z1(θ∗2, θ

∗2) = U (θ∗

2)

2κ− (1 + T ′(θ∗

2))X ′(θ∗2 + T (θ∗

2)) > 0,

where the strict inequality follows since θ∗2 ≤ θ + T (θ) (see Lemma 1) and κ ≤ κ. Since Z is con-

cave in its first argument, it follows that Z (θ, θ∗2) < 0 for all θ < θ∗

2, which contradicts Z (θ1, θ∗2) ≥ 0

in the former case, and Z (θ∗1, θ

∗2) ≥ 0 in the latter case. This completes the proof of Claim B.

We are now ready to complete the proof. By Claim B, for any ε > 0 and any P ∈ P, there existsθP,ε ∈ �P ∩ [θ − ε, θ] such that v(P, φ = θP,ε + κ) ≥ 0. As such, the integral

∫∪P∈P (�P ∩[θP,ε,θP,ε+2κ])

v(P(θ), φ = θP(θ),ε + κ)μ(dθ) (11)

is weakly positive. Since v is a conditional expectation (see its definition (9)), the integral is alsoequal to ∫

∪P∈P (�P ∩[θP,ε,θP,ε+2κ])(V (θ) − C)μ(dθ).

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The domain of the integral (11) can be expanded as

(� ∩ [θ, θ + 2κ − ε]) ∪⋃P∈P

(�P ∩ [θP,ε, θ]) ∪⋃P∈P

(�P ∩ [θ + 2κ − ε, θP,ε + 2κ]).

The term V (θ) − C is strictly negative over the first set in the domain above, with the singleexception of at θ. For all ε small enough and by Claim A, the first set has a strictly positivemeasure, while the other two have measures that approach zero. As such, the integral in expression(11) is strictly negative for ε small enough. The contradiction completes the proof.

Proof of Proposition 2 (equilibrium nonexistence when κ > T (θ − 2κ)/2). We show, by contra-diction, that there is no equilibrium if κ > T (θ − 2κ)/2. Suppose to the contrary that an equilibriumexists. Let P(·) be the equilibrium price function. From the main text, the equilibrium cannot befully revealing, and so define �∗ to be the nonempty set of fundamentals at which the price is notfully revealing, i.e.,

�∗ = {θ : ∃θ′ = θ such that P(θ) = P(θ′)}.

Given �∗, define θ∗ = inf �∗. We prove the following claims.

Claim 1. If θ < min{θ∗, θ}, then P(θ) = X (θ) + U (θ), and if θ ≥ min{θ∗, θ}, then P(θ) ≥X (min{θ∗, θ}) + U (min{θ∗, θ}).

Proof of Claim 1. By definition, if θ < θ∗, the price is fully revealing. So if θ < θ also, theagent intervenes, and P(θ) = X (θ) + U (θ). So for any θ < min{θ∗, θ}, the price is X (θ) + U (θ).Next, suppose that contrary to the claim P(θ′) < X (min{θ∗, θ}) + U (min{θ∗, θ}), for some θ′ ≥min{θ∗, θ}. But then there exists θ < min{θ∗, θ} ≤ θ∗ such that P(θ) = P(θ′), contradicting thefact that θ∗ = inf �∗. This completes the proof of Claim 1.

Claim 2. θ∗ < θ, and so T (θ∗) > 0.

Proof of Claim 2. Suppose to the contrary that θ∗ ≥ θ, so that min{θ∗, θ} = θ. By Claim 1,P(θ) = X (θ) + U (θ) if θ < θ, and P(θ) ≥ X (θ) + U (θ) for θ ≥ θ. As such, whenever the truefundamental is strictly above θ, the agent knows either that the fundamental is strictly above θ, orthat the fundamental is either strictly above θ or equal to θ, with a positive probability of both. Sothe agent intervenes with probability 0 for any θ > θ. But then the price is not above X (θ) + U (θ)for any θ close to θ. This contradiction completes the proof of Claim 2.

Claim 3. P(θ∗) = X (θ∗) + U (θ∗), and so E(I |θ∗) = 1.

Proof of Claim 3. From Claims 1 and 2, P(θ) ≥ X (θ∗) + U (θ∗) for θ ≥ θ∗. The claim followssince by Claim 2, U (θ∗) > 0 and thus P(θ∗) ≤ X (θ∗) + U (θ∗).

Claim 4. θ∗ ≥ θ − 2κ.

Proof of Claim 4. Suppose otherwise that θ∗ < θ − 2κ. Observe that X (θ − 2κ) + U (θ − 2κ) =X (θ − 2κ + T (θ − 2κ)) < X (θ). So there exists θ1 ∈ �∗ with a price P strictly below X (θ). Butany fundamental θ2 ≥ θ has a price of at least min{X (θ2), X (θ2 + T (θ2))} ≥ min{X (θ), X (θ +T (θ))} ≥ X (θ). So all fundamentals with price P lie below θ, implying that the agent interveneswith probability 1 at all of them, and hence θ1 is the unique fundamental associated with price P .But then θ1 /∈ �∗, giving a contradiction.

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Claim 5. If fundamentals θ1 and θ2 share the same price, then T (θ1) and T (θ2) have the same sign.

Proof of Claim 5. If T (θ) is everywhere positive, then the claim is vacuously true. Forthe case in which T (θ) is negative for large-enough fundamentals, define θT 0 implicitly byT (θT 0) = 0. So if θ > θT 0, we know that P(θ) ≥ X (θ) + U (θ) > X (θT 0) + U (θT 0), whilefor fundamentals θ < θT 0, we know that P(θ) ≤ X (θ) + U (θ) < X (θT 0) + U (θT 0). So it isimpossible for a fundamental to the left of θT 0 to share a price with a fundamental to the right of θT 0.

Now, consider first the case where θ∗ ∈ �∗. There exists a fundamental θ′ > θ such thatP(θ′) = X (θ′) + E(I |θ′)U (θ′) = X (θ∗) + U (θ∗). By Claim 2, T (θ∗) > 0. By Claim 5, T (θ′) > 0.Note that θ∗ > θ′ − 2κ, since if θ′ ≥ θ∗ + 2κ, the price at θ′ is at least X (θ∗ + 2κ), which since2κ > T (θ − 2κ) ≥ T (θ∗) (by Claim 4) is more than P(θ∗) = X (θ∗ + T (θ∗)).

Since E(I |θ∗) = 1, the agent always intervenes at signals below θ∗ + κ. Thus, E(I |θ′) ≥Pr(θ′ + ξ ≤ θ∗ + κ) = 1 − θ′−θ∗

2κ. Define the function Z (θ∗, θ′) as follows:

Z (θ∗, θ′) ≡ X (θ′) +(

1 − θ′ − θ∗

2κ

)U (θ′) − X (θ∗) − U (θ∗).

By the above arguments, in the proposed equilibrium, Z (θ∗, θ′) ≤ 0. We know that Z (θ′, θ′) =0, and that Z (θ′ − 2κ, θ′) = X (θ′) − X (θ′ − 2κ + T (θ′ − 2κ)) > 0. Since the security is convex,Z11 < 0. Thus, there are no θ′ and θ∗ ∈ (θ′ − 2κ, θ′) for which Z (θ∗, θ′) ≤ 0. This is a contradictionto the proposed equilibrium.

Suppose now that θ∗ /∈ �∗. There exists some sequence (θi )∞i=0 ⊂ �∗ that converges to θ∗.Moreover, by Claim 3, E(I |θi ) → 1 as i → ∞: for if this is not true, there is a θi ≥ θ∗ at whichthe price is below X (θ∗) + U (θ∗), contradicting Claim 1. For each θi in this sequence, there existsat least one fundamental, θ′

i , at which the price is the same and which lies to the right of θ.Hence, X (θ′

i ) + E(I |θ′i )U (θ′) = X (θi ) + E(I |θi )U (θi ). Note that θ′

i − θi is bounded away from 0as i → ∞ since θi → θ∗ < θ. We know that

E(I |θ′i ) =

∫ θ′i +κ

θ′i −κ

I (P(θi ), φ)1

2κdφ

≥∫ θi +κ

θ′i −κ

I (P(θi ), φ)1

2κdφ

≥(

1 − θ′i − θi

2κ

)− (1 − E(I |θi )).

Define

εi ≡ (1 − E(I |θi ))(U (θ′i ) − U (θi )),

Z (θi , θ′i ) ≡ X (θ′

i ) +(

1 − θ′i − θi

2κ

)U (θ′

i ) − X (θi ) − U (θi ),

Z (θi , θ′i ) ≡ Z (θi , θ

′i ) − εi.

By the above arguments, in the proposed equilibrium, Z (θi , θ′i ) ≤ 0. We know that εi ap-

proaches 0 (the value of intervention, U (θ), is bounded above by the maximum value of T ).We know that Z (θ′

i , θ′i ) = 0, and that Z (θ′

i − 2κ, θ′i ) = X (θ′

i ) − X (θ′i − 2κ + T (θ′

i − 2κ)) > 0.Since the security is convex, Z11 < 0. Thus, for any θi between θ′

i − 2κ and θ′i , Z (θi , θ

′i ) ≥

−εi + (θ′i −θi )(X (θ′

i )−X (θ′i −2κ+T (θ′

i −2κ)))2κ

. This implies that Z (θi , θ′i ) ≤ 0 can hold only if θ′

i −2κεi

X (θ′i )−X (θ′

i −2κ+T (θ′i −2κ))

≤ θi ≤ θ′i . Then, since εi approaches 0, there are no θ′

i and θi that are

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bounded away from each other for which Z (θi , θ′i ) ≤ 0. This is a contradiction to the proposed

equilibrium.

Proof of Proposition 3. Part (i). We first characterize in more detail the equilibria described in theproposition. There exist fundamentals θ01 < θ11 < θ and a function θ∗

2 : [θ01, θ11] → [θ, θ] withθ∗

2(θ01) = θ, such that for any set Y1 ⊂ [θ01, θ11] the following prices and intervention probabilitiesconstitute an equilibrium:

1. (Agent-preferred intervention below θ) If θ ≤ θ, the agent intervenes with probability 1, andthe price is X (θ) + U (θ).

2. (Overintervention for some θ > θ) If θ ∈ θ∗2(Y1), the agent intervenes with probability 1 −

θ−θ∗−12 (θ)2κ

> 0, and the price is X (θ) + (1 − θ−θ∗−12 (θ)2κ

)U (θ).3. (Agent-preferred intervention for some θ > θ) If θ > θ and θ /∈ θ∗

2(Y1), the agent never inter-venes, and the price is X (θ).

For use throughout the proof, define the function

Z (θ1, θ2) = X (θ2) +(

1 − θ2 − θ1

2κ

)U (θ2) − X (θ1) − U (θ1).

Intuitively, this is the difference between the price at a fundamental θ1 given an interventionprobability 1, and the price at fundamental θ2 > θ1 given an intervention probability 1 − θ2−θ1

2κ.

Observe that Z has the following properties:

Z11(θ1, θ2) < 0,

Z12(θ1, θ2) = U ′(θ2)

2κ,

Z (θ, θ) = 0,

Z (θ − 2κ, θ) = X (θ) − X (θ − 2κ + T (θ − 2κ)).

We start by establishing

Lemma 2. For κ < T (θ)/2 sufficiently close to T (θ)/2 and |T ′(θ)| sufficiently small, there exists aunique θ01 < θ such that

X (θ) +(

1 − θ − θ01

2κ

)U (θ) = X (θ01) + U (θ01).

Proof of Lemma 2. Since 2κ < T (θ) ≤ T (θ − 2κ), we know that Z (θ − 2κ, θ) < 0 andZ (θ, θ) = 0. Since Z11 < 0, the result follows provided Z1(θ, θ) < 0. We know that

Z1(θ, θ) = U (θ)

2κ− X ′(θ + T (θ))(1 + T ′(θ))

= X (θ + T (θ)) − X (θ)

2κ− X ′(θ + T (θ))(1 + T ′(θ))

= 1

2κ

∫ θ+T (θ)

θ

(X ′(θ) − 2κ

T (θ)X ′(θ + T (θ))(1 + T ′(θ))

)dθ.

Since X is a convex function, Z1(θ, θ) < 0 for all 2κ close enough to T (θ) and |T ′(θ)| sufficientlysmall.

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First observe that since Z (θ01, θ) = Z (θ, θ) = 0, and Z11 < 0, then Z (θ1, θ) > 0 for any θ1 ∈(θ01, θ). Moreover, Z (·, θ) is single-peaked. Let θ11 ∈ (θ01, θ) be its maximum. Since for anyθ1 ∈ (θ01, θ11), Z (θ1, θ) > 0 and Z (θ1, θ1 + 2κ) < 0, by continuity there exists some θ2 > θ, forwhich Z (θ1, θ2) = 0. We define a function, θ∗

2(θ1), where θ∗2 is the smallest θ2, above θ, for which

Z (θ1, θ2) = 0. Economically, θ∗2(θ1) is the fundamental which has the same market price as θ1. We

know that θ∗2(θ01) = θ.

The function θ∗2(θ1) is strictly increasing over [θ01, θ11], as follows. Note that

Z (θ1, θ2) = Z (θ1, θ) +∫ θ2

θ

Z2(θ1, y) dy.

Since Z (θ1, θ) increases over [θ01, θ11], and Z12 > 0 (provided that |T ′| is sufficiently smallthat U increases), it follows that for any θ2 ≥ θ, Z (θ1, θ2) is increasing in θ1 over [θ01, θ11]. Thus,the smallest θ2, at which Z (θ1, θ2) = 0, is strictly increasing in θ1, implying that θ∗

2(θ1) is a strictlyincreasing function.

The fundamental θ is drawn from the distribution function g(θ) and the noise in the agent’s signalis uniformly distributed. Hence, if the agent observes a price and signal consistent with θ1 and θ∗

2(θ1),

he assesses the expected benefit of intervention, net of costs, asg(θ1)V (θ1)+g(θ∗

2(θ1))V (θ∗2(θ1))

g(θ1)+g(θ∗2(θ1)) − C , and

intervenes only if this expression is positive. Since θ∗2(θ01) = θ, we know that this expression is

strictly positive at θ1 = θ01. Choose θ11 ∈ (θ01, θ11] such thatg(θ1)V (θ1)+g(θ∗

2(θ1))V (θ∗2(θ1))

g(θ1)+g(θ∗2(θ1)) − C ≥ 0

for all θ1 ∈ [θ01, θ11]. Moreover, θ∗2(·) increases over this interval (since θ11 ≤ θ11).

We have now defined the values θ01 and θ11 that were used to characterize the equilibria inthe beginning of the proof. It remains to show that there is an equilibrium of the type described.This requires showing that the prices are rational given the intervention probabilities, and thatthe intervention probabilities result from the agent’s behavior given the information in the priceand his own private signal. It is immediate that the prices specified above are rational given thecorresponding intervention probabilities. Thus, we turn to show that the intervention probabilitiesresult from the agent’s behavior. We will do this by analyzing different ranges of the fundamentalsseparately.

For a fundamental θ ≤ θ and θ /∈ Y1, the price is X (θ) + U (θ) = X (θ + T (θ)). The same pricemay be observed at the fundamental θ + T (θ). Since 2κ < T (θ) ≤ T (θ), the agent’s private signalwill indicate for sure that the fundamental is θ and not θ + T (θ). Hence, the agent will choose tointervene, generating intervention probability of 1. Note that the same price cannot be observed atany fundamental below θ + T (θ). Observing such a price at a fundamental below θ + T (θ) wouldimply that the fundamental belongs to the set θ∗

2(Y1), but this contradicts the fact that θ /∈ Y1.For a fundamental θ ≤ θ and θ ∈ Y1, the price is again X (θ) + U (θ). As before, the same price

may be observed at the fundamental θ + T (θ) without having an effect on the decision of the agentto intervene at θ, given that 2κ < T (θ) ≤ T (θ). Here, however, the same price will also be observedat the fundamental θ∗

2(θ). This is because the fundamental θ∗2(θ) ∈ θ∗

2(Y1) generates a price of

X (θ∗2(θ)) + (1 − θ∗

2(θ)−θ

2κ)U (θ∗

2(θ)), which by construction is equal to X (θ) + U (θ). (Note that thesame price will not be observed at any other fundamental in the set θ∗

2(Y1), since X (θ) + U (θ) andθ∗

2(θ) are strictly increasing in θ.) Thus, at the fundamental θ, the agent observes a price that isconsistent with both θ and θ∗

2(θ), and may observe a private signal that is also consistent with bothof them. If this happens, given the uniform distribution of noise in the agent’s signal, the agent

will intervene as long asg(θ1)V (θ1)+g(θ∗

2(θ1))V (θ∗2(θ1))

g(θ1)+g(θ∗2(θ1)) − C ≥ 0. By construction, this is true for all

θ ∈ Y1, and thus, at the fundamental θ, the agent will intervene with probability 1.For a fundamental θ > θ and θ /∈ θ∗

2(Y1), the price is X (θ). The same price may be observed at afundamental θ′ ≤ θ such that θ′ + T (θ′) = θ and also at some θ′′ > θ in θ∗

2(Y1). Since 2κ < T (θ) ≤T (θ′), the agent’s private signal at the fundamental θ will indicate for sure that the fundamental isnot θ′. Hence, the agent will know that the fundamental is above θ and will choose not to intervene,generating intervention probability of 0, as is stated in the proposition.

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Finally, for a fundamental θ > θ and θ ∈ θ∗2(Y1), the price is X (θ) + (1 − θ−θ∗−1

2 (θ)2κ

)U (θ). As

follows from the arguments above, the same price will be observed at the fundamental θ∗−12 (θ)

and also may be observed at some fundamental θ′′

> θ in θ′′

/∈ θ∗2(Y1). (As argued before, two

fundamentals in the set θ∗2(Y1) cannot have the same price.) As also follows from the arguments

above, the agent will choose to intervene if and only if his signal is consistent with both θ andθ∗−1

2 (θ) (the signal cannot be consistent with both θ∗−12 (θ) and θ

′′). Due to the uniform distribution

of noise in the agent’s signal, this generates an intervention probability of 1 − θ−θ∗−12 (θ)2κ

.

Part (ii). Suppose to the contrary that there exists an equilibrium without agent-preferred inter-vention and in which the probability of intervention for all θ > θ is 0. In this equilibrium, theremust exist some θ1 < θ such that E[I |θ1] < 1. Because θ1 < θ, it follows that there must existθ2 ∈ (θ, θ1 + 2κ) with the same price as θ1. Moreover, because E[I |θ] = 0 for all θ > θ, the funda-mental θ2 is the unique fundamental to the right of θ with the same price as θ1. So the interventionpolicy I in this equilibrium must satisfy

I (P(θ1),φ) ={

0 if φ ∈ (θ2 − κ, θ2 + κ)

1 if φ ∈ (θ1 − κ, θ1 + κ) and φ /∈ (θ2 − κ, θ2 + κ).

As such, the expected intervention probability at θ1 is

E[I |θ1] = Pr((θ1 + ξ) ∈ (θ1 − κ, θ2 − κ)) = θ2 − θ1

2κ.

Define a function

Z (θ) = X (θ1) +(

θ − θ1

2κ

)U (θ1) − X (θ).

On the one hand, observe that Z (θ2) = X (θ1) + E[I |θ1]U (θ1) − X (θ2) = 0, since by hypothesis,θ1 and θ2 have the same price. But on the other hand, Z (θ1) = 0, Z (θ1 + 2κ) = X (θ1 + T (θ1)) −X (θ1 + 2κ) > 0 since 2κ < T (θ) ≤ T (θ1), and Z is concave since X (θ) is convex. As such, thereis no value of θ ∈ [θ1, θ1 + 2κ) for which Z (θ) = 0. The resultant contradiction completes theproof.

Proof of Proposition 4. (i) Suppose the agent observes the price of securities A and B, wheresecurity A is strictly convex and security B is strictly concave. The heart of the proof is thefollowing straightforward claim:Claim. For any pair of fundamentals θ1 and θ2 = θ1, there is no probability q ∈ (0, 1) such that

Xs (θ1) + qUs (θ1) = Xs (θ2) for securities s = A, B (12)

or

Xs (θ1) + qUs (θ1) = Xs (θ2 + T (θ2)) for securities s = A, B. (13)

Proof of Claim. Observe that

Xs (θ1) + qUs (θ1) = (1 − q)Xs (θ1) + q Xs (θ1 + T (θ1))

{>

<

}Xs (θ1 + qT (θ1))

if security s is

{convexconcave

}.

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Since Xs is strictly increasing for both securities, it is immediate that neither Equation (12) nor(13) can hold.

The proof of the main result applies this claim. Consider any equilibrium, and let � be the setof fundamentals that share the same price vector as a fundamental at which intervention is notaccording to the agent’s preferred rule. Suppose that (contrary to the claimed result) the set � isnonempty. Let θ∗ be its supremum. Clearly, if θ∗ ≤ θ then for all equilibrium prices associated withfundamentals � the agent would know that the true fundamental lies below θ, and would chooseto intervene. So θ∗ > θ. Moreover, by Lemma 1, θ∗ ≤ θ + 2κ < θ + T (θ). For use below, let θ∗∗be such that θ∗∗ + T (θ∗∗) = θ∗. Note that θ∗∗ ≤ θ, since otherwise θ∗ cannot be the supremum of�. So T (θ∗∗) ≥ T (θ).

By construction, for fundamentals θ > θ∗ the agent chooses not to intervene, so P(θ) = X (θ).Therefore, for all fundamentals θ ∈ �, the equilibrium price vector satisfies P(θ) ≤ X (θ∗). Con-sider an arbitrary sequence {θi } ⊂ � such that θi → θ∗. The intervention probabilities convergeto zero along this sequence, E[I |θi ] → 0 (otherwise, the equilibrium price would strictly exceedX (θ∗) for some θi ). There are two cases to consider:

Case A. On the one hand, suppose there exist some ε > 0 and some infinite subsequence {θ j } ⊂ {θi }such that for each θ j, there is a fundamental θ′

j = θ j with the same price, and E[I |θ′j ] ∈ [ε, 1 − ε].

It follows that there is a subsequence {θk} ⊂ {θ j } such that for each θk there is a fundamental θ′k =

θk with the same price, and E[I |θ′k ] converges to q ∈ [ε, 1 − ε] as k → ∞. Since for all k

Xs (θk ) + E[I |θk ]Us (θk ) = Xs (θ′k ) + E[I |θ′

k ]Us (θ′k )

for securities s = A, B, and the left-hand side converges to Xs (θ∗), it follows that {θ′k} must converge

also to, say, θ′. Thus Xs (θ∗) = Xs (θ′) + qUs (θ′) for securities s = A, B, directly contradicting theabove claim.

Case B. On the other hand, suppose that Case A does not hold. So there exists an infinitesubsequence {θ j } ⊂ {θi } such that for each fundamental θ′

j possessing the same price as θ j ,the intervention probability E[I |θ′

j ] is either less than 1/j or greater than 1 − 1/j . It fol-lows that for j large, all fundamentals with the same price vector as θ j are close to eitherθ∗ (if the intervention probability is close to 0) or θ∗ − T (θ∗∗) (if the intervention probabil-ity is close to 1): formally, there exists some sequence ε j such that ε j → 0 and such thatθ′

j ∈ [θ∗ − T (θ∗∗) − ε j , θ∗ − T (θ∗∗) + ε j ] ∪ [θ∗ − ε j , θ

∗]. But for j large enough, θ∗ − ε j > θ,

θ∗ − T (θ∗∗) + ε j < θ, and (θ∗ − ε j ) − (θ∗ − T (θ∗∗) + ε j ) = T (θ∗∗) − 2ε j > 2κ. That is, for jlarge, if the agent observes price vector P(θ j ) and his own signal, he knows with certainty whichside of θ the fundamental lies. As such, he follows his preferred intervention rule, giving a contra-diction.

(ii) Exactly as in Proposition 2, a fully revealing equilibrium cannot exist. Suppose a nonfullyrevealing equilibrium exists. So at some set of fundamentals �∗ the prices of both the concave andconvex securities must be the same for at least two distinct fundamentals. That is, the set

�∗ ≡ {θ : ∃θ′ = θ such that Pi (θ) = Pi (θ′) for all securities i}

is nonempty. The proof in Proposition 2 applies, and gives a contradiction.

Proof of Proposition 5. First, in any equilibrium where there exist θ1 < θ2 with the same equityprice, the expected intervention probabilities E[θ1|I ] and E[θ2|I ] must differ (otherwise priceswould not be identical). Given that the probability of intervention can be directly inferred fromQ(θ), then the agent can always infer θ based on P(θ) and Q(θ). Then, the agent will choose tointervene when θ ≤ θ, and not intervene otherwise. The same is true if the equilibrium prices of theequity security are fully revealing. Thus, if there is an equilibrium, it must feature agent-preferredintervention.

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Second, we show that agent-preferred intervention is indeed an equilibrium. In such an equilib-rium, the price of the equity is X (θ + T (θ)) for θ < θ and X (θ) for θ > θ. The prediction-marketsecurity has a price of 1 for θ < θ and 0 for θ > θ. Then, independent of the agent’s signal, theagent chooses to intervene below θ and not intervene above θ. This is indeed consistent with theprices, so agent-preferred intervention is an equilibrium.

Proof of Proposition 6. Denote the size of the set of parameters in [θ − T (θ), θ] over which theagent follows his preferred intervention rule as λ− (where θ is as defined in Lemma 1), and the sizeof the set of parameters in [θ, θ + T (θ)] over which the agent follows his preferred interventionrule as λ+.

By the shape of the price function under agent-preferred intervention (see Figure 2), everyfundamental θ ∈ [θ − T (θ), θ] that exhibits agent-preferred intervention implies that the interven-tion decision at θ + T (θ) ∈ [θ, θ + T (θ)] is not agent-preferred. This is because agent-preferredintervention at both θ and θ + T (θ) implies that the two fundamentals have the same price, but thisis impossible in a commitment equilibrium. Thus, the set of fundamentals with agent-preferredintervention in [θ − T (θ), θ] cannot be greater than the set of fundamentals without agent-preferredintervention in [θ, θ + T (θ)]. That is, λ− ≤ T (θ) − λ+, which implies that λ− + λ+ ≤ T (θ). Thiscompletes the proof.

Appendix B: Interpreting the no-equilibrium result

We present a very simple trading game that formalizes the intuition that the no-equilibrium resultin our rational expectations model can be translated into a market-breakdown result in an explicittrading game.

The trading game is as follows. There is a single market maker and multiple speculators. Alltrade must take place via the market maker. Both the speculators and the market maker observe thefundamental θ. As before, the agent observes only θ + ξ. After observing the realization of θ, themarket maker sets a price at which he is willing to buy or sell any quantity desired by speculators.The market maker can also abstain from posting a price, in which case no trade takes place. If themarket maker posts a price, speculators then submit buy-and-sell orders. The agent observes theprice set by the market maker and makes an intervention decision just as before.24

Clearly, this trading game is highly stylized. Its virtue, however, is that it both replicates arational expectations equilibrium (REE) when one exists and formalizes the notion that when theagent’s information is poor, the market maker abstains from posting a price and trade breaks down.Formally,

Proposition 7. (i) Let (P(θ), I (P(θ),φ)) be a REE. Then there is an equilibrium of the tradinggame in which for all fundamentals θ and all agent signal realizations φ, the market maker postsprice P(θ), and intervention takes place with probability I (P(θ),φ). Conversely, any equilibriumof the trading game with prices posted in all fundamentals corresponds to a REE.

(ii) When κ > T (θ − 2κ)/2, there exists θ∗ ∈ (θ, θ]25 such that for any θ ∈ [θ, θ∗], there is anequilibrium of the trading game in which: the market maker posts the price X (θ + T ) and the agentintervenes when θ ≤ θ; the market maker does not post a price when θ ∈ (θ, θ + T (θ)]; the marketmaker posts the price X (θ), and the agent does not intervene when θ > θ + T (θ). The equilibriado not exhibit agent-preferred intervention policy (except for when θ = θ or θ).

Part (i) of Proposition 7 follows almost immediately from definitions. Part (ii) is most easilyillustrated when the agent has no information (i.e., κ = ∞) since this avoids the need to consider

24 If the market maker does not set a price, this too is observed by the agent.

25 Recall that θ is defined by θ + T (θ) = θ.

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off-equilibrium-path beliefs (which are dealt with in the proof). In this case, for any θ such thatθ ∈ [θ, θ + T (θ)] there is an equilibrium in which the market maker posts no price when thefundamental lies in this range, and posts a fully revealing price otherwise. The key property of thisequilibrium is that for fundamentals θ in the no-price interval, any price that the market maker couldconceivably quote would lead to losses. Specifically, in equilibrium, prices above (respectively,below) X (θ + T (θ)) reveal that the fundamental is above (respectively, below) θ and lead to nointervention (respectively, intervention). So if at fundamental θ ∈ (θ, θ + T (θ)] the marker makerposts a high price, the agent will respond by not intervening, implying that the quoted price exceedsthe fundamental value of the security. In this case, speculators short the security and the marketmaker suffers losses. Likewise, quoting a low price leaves speculators with a profitable buyingopportunity.

Several features of this equilibrium are worth commenting upon. First, the equilibrium capturesthe idea that the agent’s action is hard to predict. That is, when the fundamental is in the neighbor-hood of θ, market participants are reluctant to trade at any price, because they do not know howthe agent will react.

Second, unless θ = θ or θ + T (θ), the equilibrium does not exhibit agent-preferred intervention.To see this, simply note that since the agent has no information in the above example, he mustmake the same intervention decision for all fundamentals in the no-price range. Since the no-pricerange straddles θ, intervention is consistent with the agent’s preferred rule at some fundamentalsin this range but not others. So whatever decision the agent makes upon seeing no price, it is notaccording to his preferred intervention rule in some cases.

Third, although the fundamental is not fully revealed in equilibrium, the agent does learnsomething from the drop in volume that occurs when θ ∈ (θ, θ + T (θ)]—specifically, that thefundamental is in this interval. Indeed, in the extreme equilibria in which θ = θ or θ + T (θ) thisinformation is enough to allow the agent to intervene according to his preferred rule.

Proof of Proposition 7. (i) The first half is immediate. For the second half, it suffices to showthat in any equilibrium of the trading game with prices posted in all states, the mapping fromfundamentals to prices satisfies the REE condition (3). To see this, note that since speculators havethe same information as the market maker, if the posted price is not equal to the security’s expectedpayoff then speculators could buy (or sell) the security to make positive profits. In this case, themarket maker would make negative profits.

(ii) To complete the description of the equilibrium, let the agent’s off-equilibrium-path beliefsbe such that if he observes a signal φ and a price corresponding in equilibrium to fundamentalθ < φ − κ (respectively, θ > φ + κ), then he believes that the fundamental is φ − κ (respectively,φ + κ). Moreover, the agent’s intervention decision at fundamental θ ∈ (θ, θ + T (θ)] and signal φ

is determined by the sign of

E[V (θ′)|θ′ ∈ (θ, θ + T (θ)] ∩ [φ − κ, φ + κ]].

In the conjectured equilibrium, whenever a price is posted it perfectly reveals the fundamental. Soby construction, the agent’s intervention decision is the best response. It remains only to check thatthe market maker has no profitable deviation.

For use below, note that by construction θ ≤ θ ≤ θ + T (θ) and by assumption θ − 2κ + T (θ −2κ) < θ = θ + T (θ), implying that θ − 2κ < θ and hence 2κ > T (θ − 2κ) ≥ T (θ) for all θ ≥ θ.

Consider a realization of the fundamental θ ≤ θ. For these fundamentals, the market makerposts a price and makes zero profits. He cannot profit by not posting a price. If he posts a higherprice p > X (θ + T (θ)), then regardless of the agent’s response, the value of the security is lessthan p, and so speculators will short the security and the market maker will lose money. If heposts a lower price p < X (θ + T (θ)), then (given the beliefs specified) the agent will intervene,implying that the value of the security exceeds p and the market maker will lose money. By asimilar argument, the market maker does not have a profitable deviation if θ > θ + T (θ).

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Next, suppose θ ∈ (θ, θ + T (θ)], the no-price region. First, consider a deviation by the marketmaker in which he posts a price p > X (θ + T (θ)). Let θ′ > θ + T (θ) ≥ θ be such that X (θ′) = p.Whenever the agent observes φ ∈ [θ′ − κ, θ + κ], he believes that the fundamental is θ′ and doesnot intervene. So the intervention probability is bounded above by θ′−θ

2κ, and so the security value

is bounded above by

θ′ − θ

2κX (θ + T (θ)) +

(1 − θ′ − θ

2κ

)X (θ).

This is strictly less than the quoted price X (θ′) for all θ′ ∈ (θ, θ + 2κ], since X is concave and2κ > T (θ). Likewise, if θ′ > θ + 2κ, then X (θ′) > X (θ + 2κ) ≥ X (θ + T (θ)), and so again thequoted price must exceed the value of security. So the agent loses money from a deviation of thisform.

Second, consider a deviation by the market maker in which he posts a price p ≤ X (θ + T (θ)).Let θ′ ≤ θ be such that X (θ′ + T (θ′)) = p. So the agent believes the fundamental is θ′ ifφ ∈ [θ − κ, θ′ + κ], and φ − κ if φ ∈ (θ′ + κ, θ + κ]. Since θ′ ≤ θ ≤ θ, it follows that the agentintervenes with probability 1 if θ < θ, and with probability θ+κ−(θ−κ)

2κ= 1 − θ−θ

2κif θ ≥ θ. The

value of the security under this deviation is thus X (θ + T (θ)) if θ < θ, and

(1 − θ − θ

2κ

)X (θ + T (θ)) + θ − θ

2κX (θ)

if θ ≥ θ. In the former case, the value of the security certainly lies strictly above the quoted priceof X (θ′ + T (θ′)), causing the market maker to lose money from this deviation. The same is truefor the latter case for θ ≤ θ + T (θ) = θ and θ′ ≤ θ. Finally, by continuity, this is also the case forθ ≤ θ + T (θ), and θ′ ≤ θ for all θ sufficiently close to θ.

Finally, note that (except for when θ = θ or θ) the equilibrium does not exhibit agent-preferredintervention. To see this, fix an equilibrium and consider the agent’s action when he sees a signalφ = θ and no price. If the agent intervenes, this implies that with positive probability he intervenestoo much for some θ ∈ (θ, θ + T (θ)] to the right of θ. Likewise, if the agent does not intervene,then this implies that with positive probability he intervenes too little for some θ ∈ (θ, θ + T (θ)]to the left of θ.

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